Experiments and numerical analysis of the dynamic flow characteristics of a pump−pipeline system with entrapped air during start-up

Abstract

A pump−pipeline experiment platform is designed and built for exploring the dynamic flows in a pipe-conveying fluid with entrapped air, and a corresponding CFD simulation is executed based on the volume-of-fluid (VOF) model. The transient flow characteristics of the entrapped air in pipelines under different pump and valve working conditions are studied by experimental and numerical approaches. The experimental data and simulation results are in good agreement. Both of them show that the pressure impact can be mitigated by increasing the air volume ratio (air volume/pipe volume) or the valve opening time. Two types of the air–water interfaces, a clear interface and a broken interface, appear in the process of air pressurization to the peak pressure as the flow rate rises and the valve opening time decreases. The influence of the flow rate, air volume and valve opening time on the peak pressure and arrival time are investigated, and a corresponding equation with respect to the dimensionless parameters is proposed for the pressure prediction of a system with entrapped air during start-up.

KEYWORDS:

• dynamic flow characteristic
• entrapped air
• experiment
• pump−pipeline
• start-up
• VOF model

1. Introduction

The pipeline system is an important part of fluid transportation and energy transmission in the energy, petrochemical and other industries. A common pipeline system composed of pipes, flow control valves and pumps for conveying fluids can be specified as a pump−pipeline system. When entrapped air exists in a pump−pipeline system, the entrapped air could be impacted when the pump and valve are started, resulting in transient pressure impact and pressure fluctuation. The flow characteristics caused by entrapped air in pipes may pose a threat to the normal operation and safety of a pipeline system. Accurately predicting the dynamic flow characteristics of pump−pipeline systems with entrapped air in the start-up condition can ensure its operation stability and safety.

Pump−pipeline systems are widely employed in engineering applications. Since the fluid flow is directly related to operational performance and system safety, many scholars have paid attention to research on the fluid flow characteristics, especially the flow interaction effects between pumps, valves, pipes, etc. Nikpour et al. (2014) conducted a two-dimensional (2D) computational fluid dynamics (CFD) simulation of the water hammer in a pump−pipeline system, and compared different turbulence models according to the experiment data. Gu et al. (2021, 2022) analysed and discussed the influence of the interaction between pump and pipeline on the transient flows by experimental, theoretical and numerical approaches. Wan et al. (2018) established a numerical model based on the method of characteristics (MOC) and found that increasing the pump’s moment of inertia had positive effects on water hammer control. The above studies focus on single-phase flow, while two-phase flow is not considered.

In the pump−pipeline system, air may be trapped as air pockets in the pipe owing to the release of bubbles dissolved in the liquid, rapid injection after interruption, and other reasons. A significant pressure impact of entrapped air caused by changes of operating conditions has been found in practical pump−pipeline systems. The corresponding problem has attracted the attention of some scholars. Izquierdo et al. (1999) used a one-dimensional (1D) model to study rapid pressurization in a pump−pipeline containing multiple entrapped air pockets, and concluded that smaller air pockets would induce higher peak pressure. Fuertes et al. (1999) extended the work of Izquierdo et al. (1999) and investigated the effects of several relevant parameters on air peak pressure, including the characteristic curve shape of the pump, the slope of the pipe and friction. However, these studies only carried out numerical analysis, and research on the effects of some of the key factors, such as valve opening time, flow rate, etc., on pressure impact was relatively lacking. The existing research results are insufficient to solve the pressure impact problem caused by entrapped air in pump−pipeline systems. It is necessary to conduct systematic and in-depth research on the dynamic flow characteristics of pump−pipeline systems with entrapped air, especially during start-up.

Conversely, research on simple pipeline with entrapped air, especially reservoir pipelines, has been carried out earlier. Sudden compression of the entrapped air in reservoir pipelines has been identified as the reason for transient pressure impact (Martin, 1976; Wylie et al., 1993). It has also been confirmed by many scholars through experiments and engineering cases (Pozos-Estrada et al., 2012; Wright et al., 2012; Zhou, Liu, Karney, et al., 2011). Early numerical models for transient flows with entrapped air were usually 1D and could be divided into three categories: the rigid water column model (RWCM) (Izquierdo et al., 1999; Martin, 1976; Zhou et al., 2002), the elastic water model (EWM) (Chaudhry, 1979; Wylie et al., 1993) and the Preissmann slot method (PSM). RWCM neglects the compressibility of the fluid and the elasticity of the pipe walls, while EWM considers both. RWCM and EWM assume that the air–water interface is perpendicular to the pipeline. However, Zhang et al. (2020) provided a way to avoid this assumption by using MOC to solve the pressure flow and free surface flow. Based on PSM, Vasconcelos and Wright (2004) incorporated air phase pressure into the momentum equation to solve the pressurized flow under limited ventilation. One-dimensional numerical models are idealized and simplified, and provide limited information. Many studies have reported that 1D models may overestimate the peak pressure and misjudge the variation rule of peak pressure (Huang & Zhu, 2021; Zhou, Liu, Karney, et al., 2011). Therefore, 1D numerical models are limited as regards studying transient flows with entrapped air. The development of CFD and the volume-of-fluid (VOF) model has provided an attractive approach for further understanding the dynamic behaviour of two or more fluids. He et al. (2022) described the dynamic behaviour of an isolated slug driven by pressurized air in a voided line with an end orifice by a three-dimensional (3D) CFD model. Zhou, Liu, and Ou (2011) were the first to apply the VOF model to the simulation of transient flow in a water filled pipe containing entrapped air. Many studies have confirmed that CFD and the VOF model are more effective than 1D models in flow simulation with entrapped air in terms of physical correlation and numerical accuracy (Hargreaves et al., 2007; Martins, 2012).

The transient flows are different in reservoir−pipeline and pump−pipeline system (Malekpour, 2014). Advanced methods and rich contents are emerging in the study of reservoir−pipeline systems with constant pressure upstream, while the research about pump−pipeline systems are still relatively simple. Vasconcelos and Wright (2009) believe that the use of multi-dimensional models to simulate the formation and movement of air pockets is an important improvement for dynamic flows in pump−pipelines with air. Pozos-Estrada et al. (2016) proved that it is desirable to analyse the potential destructive effects of air pockets on hydraulic transients for various conditions of pump operation. Therefore, it is necessary to study the dynamic flows of pump−pipeline systems by CFD simulation. Unlike a constant pressure in the reservoir, the output pressure of the pump is difficult to control accurately in an experiment, and the function of the pump cannot be expressed easily in CFD. This is a major obstacle to the study of transient flows in pump−pipeline systems with entrapped air.

In this article, experiments and CFD numerical simulations are carried out to study systematically the dynamic flow of pump−pipeline systems with entrapped air during start-up. The output pressure of the pump is controlled in the experiments and CFD by using the characteristic curve of the pump. The influence is analysed of three important parameters, namely the flow rate, the entrapped air volume and the valve opening time, on the peak air pressure and the arrival time of peak pressure. A calculation equation for peak air pressure considering the above three parameters is proposed with respect to the dimensionless parameters. The movement of entrapped air is preliminarily understood through experiments and CFD, and the flow information of CFD provides a basis for further research and interpretation of air dynamics.

2. Experimentation and CFD simulation of pump−pipelines with entrapped air

An experimental system of dynamic flow with entrapped air is designed and built in order to study the transient pressure impact on the pump−pipeline system with entrapped air during start-up. Based on the experimental system and working conditions, a 2D-VOF CFD model of a pump−pipeline system with entrapped air is established so as to conduct numerical simulation analyses and combine with the experimental results to explore the action mechanism and influence of flow rate, air volume and valve opening time on the dynamic flow characteristics.

2.1. Experimental system

As shown in Figure 1, an experimental system consisting of a pump−pipeline with entrapped air is designed and built to study the effects of flow rate, air volume and valve opening time on the dynamic flow characteristics, and relevant experimental tests and data analyses are carried out.

Figure 1. Experimental system for the dynamic flow characteristics of a pump−pipeline system with entrapped air: (a) general view of the experimental system (1 − horizontal pipe with entrapped air, 2 − dead-end valves, 3 − pump and control valve, 4 − water tank, 5 − air injection system); (b) pressure sensor and dead-end valve; (c) air injection system; (d) pump and control valve; (e) control and power system.

The system is composed of a polymethyl methacrylate (PMMA) pipe with entrapped air, a dead-end valve, a pump and control valve, a water tank, an air injection system, a control and power system, a main pipe, a return pipe, and elbows, tees, etc. The main pipe, the size of which is DN100, is made of stainless steel. At the back of the pipeline, a horizontal pipe of length 2 m and a vertical pipe of length 0.4 m are the PMMA pipes with entrapped air, which can facilitate the observation of liquid level height and fluctuation behaviour. The end of the pipeline forms a dead end by closing the valve, as shown in Figures 1(a) and 1(b). The entrapped air is formed and controlled by an air compressor, an air transmission pipe, a control valve, etc. (Figures 1(a) and 1(c)). The water comes from the water tank, the maximum volume of which is 8 m³. As shown in Figures 1(a) and 1(d), the pressure source is a horizontal centrifugal pump with a frequency conversion function, which could be used to achieve different flow rates. The control valve, located behind the centrifugal pump, is a quarter-ring butterfly valve that opens at a constant speed, and the opening time is adjustable. The whole device is controlled through the control and power system shown in Figure 1(e).

The specific composition and operation process of the experimental system are shown in Figure 2. Before the experiment, the pipe is filled with water and air is injected into the pipe by the air compressor to form air entrapped in the PMMA pipe. The volume of air can be obtained by measuring the height of the air pocket, and adjusted through the dead-end valves and pores. When the air volume reaches the set requirements, the valve on the air transmission pipe is closed to block the fluid channel between the air injection system and the main pipe.

Figure 2. Schematic of experimental system.

During the experiment, the flow rate of the centrifugal pump and the opening time of the control valve are set according to the working conditions. The centrifugal pump starts when the control valve is closed and runs at the set rotation speed. With the opening of the valve, the water in the tank is injected into the main pipe through the centrifugal pump to form the fluid impact pressure for the entrapped air, and then flows back to the tank through the return pipe. Measuring instruments include flowmeter, dynamic pressure sensor, dynamic signal analyser and camera. The flow rate is measured at a frequency of 50 Hz by an electromagnetic flowmeter with an accuracy of 0.2%. As shown in Figure 1(b), the dynamic pressure sensor (PCB 112A22, with a linearity of 0.08% FS) is located at the end of PMMA pipe, 0.1 m away from the dead end. The dynamic pressure is collected at a sampling frequency of 40 kHz. Pressure signals are recorded and processed using a dynamic signal analyser (CRYSTAL CoCo-80X). As shown in Figure 2, a camera is used to record the change of flow state in the whole horizontal PMMA pipe during the experiment. The camera allows a frame rate of 30 fps for a 3840 × 2160 pixels resolution.

In the experiment, the dynamic flow characteristics are studied under 24 working conditions, composed of two kinds of flow rate (3 m³·h⁻¹ and 6 m³·h⁻¹), four kinds of air volume (0.66 × 10⁻³, 3.6 × 10⁻³, 6.7 × 10⁻³ and 11 × 10⁻³ m³), and three kinds of control valve opening time (1, 2.5 and 5 s).

2.2. CFD model and numerical simulation

According to the designed experimental system and experiment conditions, a CFD model of the pump−pipeline with entrapped air can be established using Fluent^® software and the 2D-VOF model so as to carry out a numerical simulation analysis of the transient pressure impact.

2.2.1. Flow path of pump−pipeline system

In the pump−pipeline system, the water in the tank is injected into the main pipe through the centrifugal pump and control valve, forming a fluid impact pressure for the entrapped air, and then flows back to the tank through the return pipe. According to the flow process of the system, the water and entrapped air in the pipe downstream of the control valve can be selected as a fluid domain for numerical simulation, while the water tank, centrifugal pump and control valve constitute the upstream boundary, as shown in Figure 3, where the relevant physical parameters include the potential head of water level in the tank, H_R, the centrifugal pump head, H_P, the loss coefficient of the control valve, k, the water velocity at the inlet of the fluid domain, v, the water pressure at the inlet of the fluid domain, P_w, the potential head at the location of entrapped air, H_T, and the pressure of entrapped air, P. Thus, the 2D geometric model corresponding to the fluid region in Figure 3 can be established for meshing. It consists of an 8.6 m long horizontal pipe, a 2 m long horizontal PMMA pipe and a vertical connection pipe, as well as an outlet pipe. Considering the flow area consistency of the 2D model and an actual 3D pipe, the diameter of the main pipe in the 2D model is equivalent to 95 mm and that of the outlet pipe is 10.78 mm.

Figure 3. Flow path of the pump−pipeline system. (air 

water 

.)

2.2.2. 2D CFD model

Two mesh parameters, namely the global mesh parameter X and the horizontal PMMA pipe mesh parameter Y, are set with three different sizes, respectively, to ensure the accuracy and efficiency of the numerical simulation. The numerical simulation is carried out under the same conditions (flow rate 3 m³·h⁻¹, air volume 6.7 × 10⁻³ m³, valve opening time 1 s) after corresponding mesh generation. The number of mesh nodes and the maximum peak pressure, P_max, obtained by numerical simulation are listed in Table 1 in ascending order according to the number of nodes. In Table 1, i is the number of the mesh parameter combination; $P_{Smax}^i$ is the peak pressure under each i in the simulation; $P_{Emax}^{}$ is the peak pressure in the experiment; $E_1=|P_{S\text{max}}^i-P_{S\text{max}}^9|/P_{S\text{max}}^9\times 100\text{\%}$;$E_2=|P_{S\text{max}}^i-P_{E\text{max}}^{}|/P_{E\text{max}}^{}\times 100\text{\%}$. When X = 5 mm and Y = 2 mm, E₁ and E₂ are small enough to prove the accuracy of the simulation, and there is no significant change in E₁ and E₂ as the node number increases. This indicates that the simulation data are in good agreement with the experimental results, and that further refinement of the mesh size causes little improvement in the result, but it increases the calculation time significantly. Therefore, a mesh size of X = 5 mm and Y = 2 mm is selected to generate the mesh and form the 2D CFD model as shown in Figure 4.

Figure 4. CFD mesh. (air 

water 

.)

Table 1. Mesh independence analysis.

				Peak pressure (kPa)
i	X (mm)	Y (mm)	Node number	$P_{S\text{max}}^i$	$P_{E\text{max}}^{}$	$E_1\mathrm{\%}$	$E_2\mathrm{\%}$
1	10	5	42,892	24.15	19.96	14.78	20.97
2	5	5	76,996	23.43		11.36	17.36
3	10	2	95,427	22.36		6.27	12.00
4	5	2	128,329	21.30		1.24	6.69
5	10	1	257,390	21.22		0.86	6.29
6	5	1	291,867	21.19		0.71	6.14
7	2	5	300,935	21.17		0.62	6.04
8	2	2	336,184	21.11		0.33	5.74
9	2	1	495,263	21.04		–	5.39

As shown in Figure 4, the entrapped air is located in the horizontal PMMA pipe and the pressure monitoring point is set at its end. The VOF model provided in Fluent software can simulate two or more immiscible fluids with recognisable interfaces by defining the percentage of each phase in each element. Thus, The VOF model is used to analyse the interaction between air and water in the PMMA pipe.

2.2.3. Boundary conditions and settings of CFD

The inlet boundary of the numerical fluid domain can be defined as the pressure inlet, as shown in Figure 3, which is related to the water tank, centrifugal pump and control valve, then (1) $H_R+H_P-k\frac{v^2}{2g}=\frac{P_w}{\rho g}+\frac{v^2}{2g}$(1) where ρ is the density of fluid and g is the acceleration due to gravity. The Suter curve of the centrifugal pump is applied to describe the general pump behaviour. According to the Suter curve, (2) $H_P=A_P-C_P{Q}^2=A_P-C_P{S}^2{v}^2$(2) where A_P is the closing head of the centrifugal pump, C_P is a constant defining the characteristic curve of the centrifugal pump, Q is the flow rate of the centrifugal pump, and S is the cross-sectional area of the pipe. By substituting Equation (2) into Equation (1), (3) $\frac{P_w}{\rho g}=H_R+A_P-\left(g{C}_P{S}^2+\frac{k+1}{2}\right)\frac{v^2}{g}$(3)

According to the study of Izquierdo et al. (1999), the value of k is (4) $\{\begin{array}{ll}k=k_0(t/T{)}^{-1.6},& 0<t<T\\ k=k_0,& t\ge T\end{array}$(4) where k₀ is the loss coefficient when the valve is fully opened, t is the time, and T is the opening time of the control valve. Equations (3) and (4) describe the relationship between P_w and v when the centrifugal pump operates and the control valve is started. Based on Equations (3) and (4), the inlet boundary of the CFD model is written by the user defined function (UDF) of Fluent and the C language. The main process of UDF writing isas follows.

1. At the end of each time step of numerical simulation, perform the following steps: (a) obtain the total area of the inlet elements and the sum of the velocities of each element; (b) obtain the average velocity weighted by the area of the inlet elements; (c) assign the average velocity to v in Equation (3) and find P_w.

2. At the beginning of the next time step of the numerical simulation, assign P_w to the pressure inlet boundary.

The outlet boundary of the 2D CFD model is at constant pressure. A realizable $k-ϵ$ model is used for turbulence. Air is regarded as an ideal gas and water is regarded as a compressible fluid. The surface tension coefficient is set to 0.072 N·m⁻¹. Considering the roughness of the pipe wall, the wall roughness height of the stainless steel pipe is set to 0.046 mm, and that of the PMMA pipe is set to 0.0015 mm. The boundary conditions at the pipe wall are described by a ‘no slip condition’. The flow solver uses the semi-implicit method for pressure-linked equations. Convergence can be evaluated by progressively tracking the residuals at each iteration step. When the residual is less than 10⁻³, i.e. the convergence standard preset by the iterative solver, the convergence solution is realized. The time step is set according to the Courant number, which is defined as (5) $C_N=\frac{u\mathrm{\Delta }t}{\mathrm{\Delta }x}$(5) where C_N is the Courant number, u is the fluid velocity, Δt is the time step and Δx is the mesh element size. To ensure convergence of numerical simulation, C_N is set to one, then the time step is Δt = Δx/u. The time step ranges from 10⁻⁵∼5 × 10⁻⁴ s depending on the different pressure inlets.

According to the principle of equivalent air volume ratio (air volume/pipe volume) with the experiment pipeline, CFD simulations are carried out for four air volume ratios, 1.12%, 4.93%, 10.00% and 16.72%, as shown in Table 2. In addition to the four air volume ratios, the simulations also consider three steady flow rates (3, 6 and 8.5 m³·h⁻¹) and six valve opening times (0.1, 0.2, 0.5, 1, 2.5 and 5 s) in corresponding combinations, to form a total of 72 kinds of simulation conditions.

Table 2. Initial air volume ratios of the CFD model and experiment.

Note. The region shown in the table is the blue area recorded by the camera in Figure 2.

3. Analysis and discussion of the dynamic flow characteristics of the pump−pipeline system with entrapped air

Based on the above experiments and numerical simulations, the experiment data and CFD simulation results of the transient pressure and flow state under different working conditions are obtained and analysed so as to discuss the dynamic flow characteristics and flow state of the pump−pipeline system with entrapped air.

3.1. Dimensionless parameters

In order to extrapolate the analysis results to general pump−pipeline systems with different hydraulic conditions, the following four dimensionless parameters are defined.

The first dimensionless parameter, λ, was proposed by Martins et al. (2017). It is described by the ratio of wave velocity to the theoretical Torricelli velocity caused by the initial pressure difference. In this research, it is used to characterise the steady flow rate of the system, which can be achieved by adjusting the rotatation speed of the centrifugal pump under a constant pipeline system configuration and parameters. (6) $\begin{array}{rl}\lambda & =\frac{c}{\sqrt{2\left(P_s+\rho g{C}_P{Q_s}^2-P_0\right)/\rho }}\\ & =\frac{c}{\sqrt{2g(H_R+A_P-H_T)-2P_0/\rho }}\end{array}$(6) where c is the wave velocity, P_s is the steady pressure of the entrapped air, Q_s is the steady flow rate of the system, P₀ is the initial pressure of the entrapped air, and V = [2(P_s+ρgC_pQ_s²-P₀)/ρ]^0.5 is obtained by Torricelli's law. When the pipeline system is running stably, the entrapped air is stationary, as shown in Figure 3: (7) $\begin{array}{rl}{P}_s& =\rho g\left[H_R+{\left(H_P\right)}_s-H_T\right]\\ & =\rho g\left(H_R+A_P-C_P{Q}_s^2-H_T\right)\end{array}$(7) where (H_P)_s is the steady head of the centrifugal pump. Since A_P and Q_s could vary with the rotation speed of the centrifugal pump, λ can be used to characterise the steady flow rate of the system.

The second dimensionless parameter, α, is used to characterise the entrapped air volume, which is described by the ratio of the initial air volume to the total volume: (8) $\alpha =\frac{V_{\text{air}0}}{V_{\text{air}0}+V_{\text{water}0}}\times 100\text{\%}$(8) where V_{air 0} is the initial volume of entrapped air and V_{water 0} is the initial volume of water in the closed pipe section.

The third dimensionless parameter, τ, represents the opening time of the control valve. It is described by the ratio of the opening time to the time required for the pressure wave to travel from the control valve to the pressure monitoring point: (9) $\tau =\frac{T}{l/c}$(9) where l is the axial distance of the pipeline from the control valve to the pressure monitoring point.

The fourth dimensionless parameter, h_m, is the maximum transient overpressure, which is used to describe the extent to which the maximum peak pressure exceeds the steady pressure: (10) $h_m=\frac{P_{max}-P_s}{P_s-P_0}$(10)

Dimensionless parameters corresponding to parameters in different working conditions are shown in Table 3.

Table 3. Working condition parameters and corresponding dimensionless parameters for the numerical simulation.

(a) Volumetric flow rate
Qs (m^3·h^−1)	3	6	8.5
λ	19.07	9.53	7.01
(b) Air volume
Vair 0 (×10^−3 m^3)	0.66	3.6	6.7	11
α (%)	1.12	4.92	10.00	16.72
(c) Valve opening time
T (s)	0.1	0.2	0.5	1	2.5	5
τ	2	4	10	20	50	100

3.2. Variable inlet pressure

Based on the CFD model, numerical simulations are carried out in various working conditions. The inlet pressure of different working conditions is given by Equations (3) and (4). The inlet pressure curves of some working conditions are shown in Figure 5. By comparing the four curves in Figure 5, it can be seen that λ, α and τ all have an influence on the inlet pressure curves. Different λ could lead to different peak and steady values of the inlet pressure curves, while different τ and α could result in different peak pressures and arrival times to peak pressure, and basically the same steady value of the inlet pressure curves. It can be seen that λ directly affects the inlet pressure through A_P from Equations (3) and (6). According to Equations (4) and (9), the inlet pressure curves in the opening process of the control valve could change if τ changes, thus affecting the peak pressure and arrival time. However, k₀ is a constant, so the steady values of the inlet pressure curve under different τ are basically the same. The influence of α on the inlet pressure is not directly set by the equations like λ and τ, but α indirectly affects the inlet pressure mainly by influencing the fluctuation period of the entrapped air. As can be seen from Equation (3), the inlet pressure is affected by the fluid velocity, v. Different α could lead to different fluctuation periods of entrapped air, and different fluid velocity v at the same t, thus affecting the inlet pressure. In summary, the transient pressure impact and pressure fluctuation of the pump−pipeline system with entrapped air during start-up is related to the interaction between the variable pressure associated with the pump-valve control and the dynamic characteristics of the entrapped air. The dynamic flow characteristics are affected by the pump flow rate, the opening time of the control valve and the air volume ratio.

Figure 5. Inlet pressure curves in the CFD simulation under different working conditions.

3.3. Analysis of the dynamic flow characteristics

Based on the CFD model, the dynamic flow characteristics of the pipeline with entrapped air are numerically calculated under multiple working conditions, and pressure curves at the monitoring point under different dimensionless parameters λ, α and τ are obtained, as shown in Figures 6, 7 and 8, in which the pressure curves obtained from RWCM and the fluctuating pressure obtained by experiment are also given. Note that the fluctuating pressure refers to the fluctuating component of the pressure, and the direct current component is excluded.

Figure 6. Pressure curves under different λ (α = 16.72%; τ = 50): (a) λ = 19.07; (b) λ = 9.53.

It is worth noting that pressure fluctuation exists throughout the pipeline during transient events. We set several monitoring points in the experiment and simulation, but it is found that the pressure fluctuation at the monitoring point in Figures 2, 3 and 4 is the largest and most threatening to operational safety. Therefore, only the pressure fluctuation at this monitoring point is described and discussed for the sake of concise description.

As shown in Figures 6, 7 and 8, the pressure curves of RWCM are obviously inconsistent with those of experiment and  the CFD model, and the average value of the peak error is 12.25%. Therefore, as described in the introduction, the RWCM has limitations for solving transient flows with entrapped air in horizontal pipes. The pressure curves obtained by numerical simulation are basically consistent with those obtained by experiment, and the main features of the pressure curves, such as the peak pressure and the arrival time to the peak pressure (T_p), are largely consistent. Tables 4 and 5 show h_m and T_p obtained from CFD simulation and experiment under all experimental conditions, as well as the errors between simulation and experiment, respectively. It can be seen from Tables 4 and 5 that the errors in h_m and T_p are below 10% in most working conditions. Therefore, the CFD model established in this research can simulate effectively the dynamic flow characteristics of pump−pipeline systems with entrapped air during start-up.

Table 4. CFD simulation h_m, experimental results and error.

			α (%)
λ	τ	Type	1.12	4.92	10	16.72
19.07	20	Experiment	1.302	0.915	0.664	0.560
		Simulation	1.431	0.950	0.705	0.587
		Error (%)	9.887	3.804	6.151	4.769
	50	Experiment	0.880	0.614	0.576	0.502
		Simulation	1.001	0.667	0.557	0.484
		Error (%)	13.789	8.607	3.323	3.527
	100	Experiment	0.520	0.397	0.342	0.335
		Simulation	0.576	0.425	0.362	0.321
		Error (%)	10.814	7.210	5.857	4.018
9.53	20	Experiment	0.845	0.641	0.406	0.306
		Simulation	0.945	0.590	0.382	0.304
		Error (%)	11.862	7.998	5.808	0.657
	50	Experiment	0.581	0.412	0.272	0.221
		Simulation	0.558	0.375	0.249	0.204
		Error (%)	3.939	9.156	8.496	7.833
	100	Experiment	0.146	0.123	0.084	0.065
		Simulation	0.156	0.117	0.080	0.062
		Error (%)	7.036	5.052	4.656	5.134

Table 5. T_p of the CFD simulation, experimental results and the error.

			α
λ	τ	Type	1.12%	4.92%	10%	16.72%
19.07	20	Experiment(s)	0.200	0.209	0.325	0.391
		Simulation(s)	0.176	0.240	0.328	0.374
		Error (%)	12.000	14.921	0.859	4.257
	50	Experiment(s)	0.200	0.215	0.342	0.410
		Simulation(s)	0.187	0.244	0.357	0.406
		Error (%)	6.500	13.511	4.447	1.014
	100	Experiment(s)	0.202	0.275	0.381	0.447
		Simulation(s)	0.196	0.290	0.381	0.462
		Error (%)	2.970	5.455	0.041	3.416
9.53	20	Experiment(s)	0.153	0.205	0.302	0.369
		Simulation(s)	0.149	0.239	0.330	0.387
		Error (%)	2.066	16.296	9.049	5.055
	50	Experiment(s)	0.190	0.257	0.384	0.425
		Simulation(s)	0.174	0.271	0.373	0.454
		Error (%)	8.421	5.316	2.942	6.874
	100	Experiment(s)	0.200	0.291	0.422	0.475
		Simulation(s)	0.188	0.284	0.423	0.501
		Error (%)	5.900	2.385	0.223	5.455

According to Figure 6 and Table 4, the peak pressure increases while h_m decreases when λ decreases. That is, larger steady flow rates lead to higher peak pressures, but lower maximum overpressure. A larger steady flow rate requires a larger pump speed and head, leading to a higher inlet pressure of the fluid domain, thus increasing the peak pressure and steady pressure. However, the increase of the steady pressure is larger than that of the peak pressure, which is manifested as a decrease of the maximum overpressure under the working conditions studied in this research.

Figure 7 and Table 4 show that the peak pressure and h_m decrease when α increases. This implies that a larger air volume ratio leads to a lower peak pressure and a maximum overpressure, which is consistent with the studies of Izquierdo et al. (1999) and Malekpour (2014). When the initial pressure and steady pressure of air remain constant, a larger air volume could result in a larger variation of air volume and a longer T_p. As the air pressure reaches its peak, water entering the system all flows to the outlet pipe, where the fluid velocity increases with t. Thus, a larger T_p could result in a larger v at T_p and a lower inlet pressure, leading to a lower peak pressure and a maximum overpressure.

Figure 7. Pressure curves under different α (λ = 19.07; τ = 50): (a) α = 1.12%; (b) α = 4.92%; (c) α = 10%; (d) α = 16.72%.

According to Figure 8 and Table 4, the peak pressure and h_m decrease when τ increases. This means that a longer valve opening time could lead to a lower peak pressure and maximum overpressure. According to Equations (3) and (4), when the valve is opened for a longer time, k at the same t in the opening process of the control valve is larger. As a result, the inlet pressure of the fluid domain is lower, leading to a lower peak pressure and a maximum overpressure.

Figure 8. Pressure curves under different τ (λ = 19.07; α = 16.72%): (a) τ = 20; (b) τ = 50; (c) τ = 100.

The influence of λ on T_p is not significant, as can be seen from Figure 5. As the steady flow rate increases, the inlet pressure increases, resulting in a larger fluid velocity and a larger air volume change. A larger fluid velocity reduces T_p, while a larger air volume change increases T_p. Thus, the effect of λ on T_p is not significant under the working conditions studied in this research. T_p tends to increase with increasing α, i.e. a larger air volume ratio could result in a longer arrival time to peak pressure. This trend can be attributed to Boyle's law, which states that, when the initial pressure and steady pressure remain unchanged, a larger air volume could result in a larger variation of air volume and a longer T_p. T_p increases with increasing τ; namely, a longer valve opening time could result in a longer arrival time. When the valve is opened for a longer time, the k at the same t in the opening process of the control valve is larger, leading to a smaller v. As a result, T_p becomes larger with the variation of air volume unchanged.

The simulation capability of the 2D CFD approach is verified based on the above discussion. The characteristic analysis of the pressure impact can provide support for fault analysis and risk assessment of pump−pipeline systems.

3.4. Analysis of the two-phase flow state in the pipe with entrapped air

In order to understand the transient flow state of the two-phase fluid when the pump−pipeline is started, changes of the air–water flow state in the horizontal PMMA pipe are photographed by camera during the experiment, and the corresponding transient flow state is calculated by the CFD model. By analysing the changes of flow state shown in photos under various working conditions, the transient flow changes can be divided into two types. Type I is shown in Figure 9, and its typical working condition parameters are: λ = 19.07, α = 10%, τ = 20; Type II is shown in Figure 10, and its typical working condition parameters are: λ = 9.53; α = 10%; τ = 20. Both types of transient flow processes are described by photos and simulation results. In Figures 9 and 10, the region is the green area recorded by the camera in Figure 2. The left-hand side is connected to the upstream pipe, and the right-hand side is the dead end.

Figure 9. The transient flow state changes of Type I in the experiment and the CFD simulation (λ = 19.07; α = 10%; τ = 20). (air 

water

.)

Figure 10. The transient flow state changes of Type II in the experiment and the CFD simulation (λ = 9.53; α = 10%; τ = 20). (air 

water

.)

According to Figures 9 and 10, the overall change process of the two types can be described as follows.

1. t = t₁ (t₁′). After the control valve is opened, the water in the upstream pipe is impacted and enters the horizontal PMMA pipe along the bend, causing the water surface on the left-hand side to rise. The air–water interface of t₁ is clear, while the left air–water interface in the experimental photo of t₁′ is broken, and that of the simulation results is clear. The air is compressed and the air pressure begins to increase.

2. t = t₂∼t₄ (t₂′∼t₄′). The water in the upstream pipe continues to enter the horizontal PMMA pipe through the bend, pushing the surging water to occupy the entire section of the left-hand side, and forming the first surge, which is constantly moving forward. The air–water interface of t₂∼t₄ is clear, but the air–water interface on the left-hand side of t₂′∼t₄′ is broken. The air–water mixing phenomenon appears in the experimental photos and simulation results of t₂′∼t₄′. The air is continuously compressed, and the air pressure gradually increases, reaching its peak when t = t₄ (t₄′).

3. t = t₅∼t₆ (t₅′∼t₆′). The first surge moves forward, and its height is reduced by gravity and air pressure, and the water continues to form subsequent surges. The air–water interface tends to be completely clear. The water experiences an air intrusion at the pipe crown, and the air pressure decreases.

4. t = t₇ (t₇′). The first surge hits the dead end, and the water is kicked up by inertia, hitting the pipe crown. The air pressure is basically steady. The water in the upstream pipe flows to the outlet pipe and no longer enters the PMMA pipe.

5. t = t₈ (t₈′). After the first surge hits the dead end, it reverses direction and moves in the opposite direction as the subsequent surge.

6. t = t₉ (t₉′). When the two surges collide, the water arches and impacts the pipe crown. The air–water interface is partially broken and droplets splash.

7. t = t₁₀ (t₁₀′). The two surges are separated and run away from each other. The left-hand surge hits the subsequent surge, and the right-hand surge hits the dead end. The above impact process is repeated until the air–water interface is stationary.

By comparing the experimental photos and the CFD model results at different time points, the CFD model results are basically consistent with the experimental photos when the air–water interface is clear. When the air–water interface is broken (t₁′∼t₄′) in the experiment photos, the CFD simulation results can show part of the air–water mixing phenomenon, but it still tends to be clear air–water interface. It is not completely consistent with the experimental photos, which is caused by the limitations of the 2D VOF model and simulation conditions, but the CFD model can still in general display the subsequent changes of the air–water interface (t₅′∼t₁₀′). Hence the 2D-VOF CFD model can generally simulate the transient flow changes of air and water when the pump−pipeline system with entrapped air is started.

The time points of the experimental photos and simulation results in Figures 9 and 10 are marked on the air pressure curves under the corresponding working conditions, as shown in Figure 11. It is found that t₁∼t₄ (t₁′∼t₄′) is the process in which water continuously compresses air, making the air pressure gradually rises to its peak. t₅∼t₁₀ (t₅′∼t₁₀′) is the process in which water no longer enters the horizontal PMMA pipe, and the air pressure gradually becomes steady. t₁∼t₄ (t₁′∼t₄′) is the key process for studying the peak pressure and peak time of entrapped air. The main difference between the two types is in this process.

Figure 11. Pressure curves in the pipe with entrapped air under two typical working conditions.

By comparing the experimental photos for t₁∼t₄ and t₁′∼t₄′, the air–water interface on the left-hand side remains clear and steady in the process of t₁∼t₄, and there is no obvious air–water mixing phenomenon; while in the process of t₁′∼t₄′, the air–water interface on the left-hand side is partly broken, with droplets splashing and bubbles entering the water, and there is an obvious air–water mixing phenomenon. By comparing the working condition parameters of the two types, it can be seen that the λ of Type II is smaller. The flow rate and the upstream pressure are larger when the valve is opened, and the water flowing into the horizontal PMMA pipe is more likely to break through the air–water interface. By comparing the experimental photos under other working conditions, it is found that τ also affects the stability of the air–water interface. A smaller τ makes the water flow into the horizontal PMMA pipe and break the air–water interface more easily. α has no significant effect on the stability of the air–water interface.

The movement of entrapped air is basically understood through experiments and CFD based on the above analysis. A more intuitive interpretation of the transient pressure impact with entrapped air is understood by connecting the movement of the air–water interface with pressure information. The flow field information of CFD provides the basis for further study and interpretation of the air motion behaviour.

4. Calculation equation of the peak pressure in the pump−pipeline system with entrapped air

On the basis of the above study of the influence of flow rate, air volume and valve opening time on peak pressure, through the statistical analysis of a large number of simulation results, a calculation equation for the maximum overpressure h_m in terms of λ, α and τ can be obtained: (11) $\begin{array}{rl}{h}_m& =-1.91\times {10}^{-5}(\lambda -0.567)({\lambda }^{-0.829}\ln \alpha \\ & -0.249\ln \lambda +0.723\left)\right(A\times \lambda +B)\end{array}$(11) where A and B are functions of α and τ. A = 5ατ+385α+5τ−187, and B = 2ατ+11,224α+176τ−15,329.

Equation (11) is the equation of peak pressure based on the simulation results of the CFD model, and describes the quantitative relationship between the dimensionless parameters λ, α, τ and h_m. The quantitative effects of steady flow rate, air volume and valve opening time on peak pressure can be analysed by this equation. Equation (11) can be used to calculate the peak pressure of a pump−pipeline system with entrapped air during start-up. Figure 12 shows the curve of h_m with the parameters λ, α, and τ according to Equation (11), as well as the CFD simulation results.

Figure 12. Comparison of the CFD model and the calculation equation presented in this research: (a) λ = 19.07; (b) λ = 9.53; (c) λ = 7.01.

Table 6 shows the errors between Equation (11) and the experimental data. It can be seen that the errors are mostly less than 10%, which reflects the consistency of the overall data. It can also be seen that the average error is the largest when α = 1.12%, which we believe is caused by limitations of the 2D model. This formula is more suitable for the working condition where α is greater than 1.12%. When α is less than 1.12%, which is closer to the water hammer, a conservative conclusion can be obtained by using the corresponding single-phase water hammer calculation formula.

Table 6. Error between the experiment data and the calculation equation presented in this research (%).

		α
λ	τ	1.12%	4.92%	10.00%	16.72%
19.07	20	5.84	1.46	2.17	2.19
	50	4.65	1.28	2.09	3.19
	100	6.74	10.98	4.67	4.49
9.53	20	15.12	0.75	0.97	5.37
	50	8.31	5.57	0.96	4.28
	100	0.35	1.79	2.13	1.60

5. Conclusions and improvements

When the pump−pipeline system with entrapped air is started, the entrapped air could induce pressure impact and fluctuation, which may pose a threat to pipeline safety. In this research, an experimental platform and a 2D-VOF CFD model are established, and the transient characteristics of a pump−pipeline with entrapped air during start-up are studied by experimental and numerical approaches. The main conclusions are as follows:

1. The CFD simulation and experimental results have a good consistency in both the pressure curves and the flow state, which verifies the rationality of the 2D-VOF CFD model used to simulate the dynamic flow characteristics of a pump−pipeline system with entrapped air during start-up.

2. The transient pressure impact and pressure fluctuation during pump−pipeline start-up are related to the interaction between the variable pressure conditions associated with the pump-valve control and the dynamic characteristics of the entrapped air. Transient pressure characteristics are affected by flow rate, control valve opening time and air volume ratio.

3. Lower peak pressure can be achieved at a smaller flow rate, larger air volume ratio and longer valve opening time; longer arrival time can be achieved at larger air volume ratio and longer valve opening time.

4. Two types of air–water interface, a clear interface and a broken interface, exist in the process of air pressurization to the peak pressure. As the flow rate rises and the valve opening time decreases, the air–water interface becomes more fragile.

5. Based on the results of the 2D-VOF CFD model, a calculation equation with respect to the dimensionless parameters is proposed for prediction of the dynamic characteristics of the pump−pipeline system with entrapped air during start-up. The formula is more suitable for working conditions where the air volume ratio is greater than 1.12%.

Accurate pressure results and abundant flow information are obtained through experiments and simulations in this article, which promotes the study of transient flow in a pump−pipeline system with entrapped air. Some work can be done to advance the research. Based on the motion phenomena of the air–water interface obtained in experiments and simulations, the dynamic of the air–water interface can be studied further and explained. The transient flows under variable pressure of more complex and longer pipeline systems with entrapped air could be studied, and a more accurate and efficient calculation method could be explored by comparing and combining the VOF model with 1D models.

Notation

A	=	function (−)
Ap	=	closing head of centrifugal pump (m)
B	=	function (−)
c	=	wave velocity (m·s−1)
CN	=	Courant number (−)
CP	=	characteristic curve constant of the centrifugal pump (−)
g	=	acceleration due to gravity (m·s-²)
hm	=	dimensionless maximum transient overpressure (−)
Hp	=	head of centrifugal pump (m)
(Hp)s	=	steady head of centrifugal pump (m)
HR	=	potential head of water level in the tank (m)
HT	=	potential head of water level at the location of entrapped air (m)
i	=	the number of the mesh parameter combination
k	=	loss coefficient of the control valve (−)
k0	=	loss coefficient of the fully opened control valve (−)
l	=	axial distance of the pipeline from the control valve to the pressure monitoring point (m)
P	=	pressure of entrapped air (kPa)
P0	=	initial pressure of entrapped air (kPa)
$P_{Emax}^{}$	=	peak pressure in the experiment (kPa)
Pmax	=	maximum peak pressure of entrapped air (kPa)
Ps	=	steady pressure of entrapped air (kPa)
$P_{Smax}^i$	=	the peak pressure under each i in the simulation (kPa)
E1	=	the error of $P_{S\text{max}}^i$ relative to $P_{S\text{max}}^9$
E2	=	the error of $P_{S\text{max}}^i$ relative to $P_{E\text{max}}^{}$
Pw	=	pressure at the inlet of the fluid domain (kPa)
Q	=	flow rate (m3·s−1)
Qs	=	steady flow rate (m3·s−1)
S	=	cross-sectional area of the pipe (m2)
t	=	time (s)
T	=	opening time of the control valve (s)
Tp	=	arrival time to the peak pressure (s)
u	=	fluid velocity (m·s−1)
v	=	fluid velocity at the inlet of the fluid domain (m·s−1)
V	=	theoretical Torricelli velocity (m·s−1)
Vair0	=	initial volume of entrapped air (m3)
Vwater0	=	initial volume of water in the closed pipe section (m3)
X	=	global mesh parameter (mm)
Y	=	horizontal PMMA pipe mesh parameter (mm)
Δt	=	time step (s)
Δx	=	mesh element size (mm)
α	=	dimensionless entrapped air volume (−)
ρ	=	density of water (kg·m−3)
λ	=	dimensionless steady flow rate (−)
τ	=	dimensionless opening time of the control valve (−)

Acknowledgements

The authors thank China Nuclear Power Engineering Co. Ltd for their help with the experiment.

Disclosure statement

No potential conflict of interest was reported by the authors.