Valve closure based on pump runaway characteristics in long distance pressurized systems

In order to guarantee the safety of pumps, valves are installed at the outlet of each pump in long-distance pressurized water supply systems. However, water hammer pressure caused by improper valve closure can tremendously exceed the standard of the pipeline. In this paper, the effects of valve closure on speed of pump and pressure along the pipeline were investigated. Valve closure formula based on pump runaway characteristics were proposed and veri ﬁ ed using numerical simulation. In addition, when valves were closed under the formula proposed in this paper and other closure laws, the minimum speed, minimum and maximum pressure along the pipeline were compared. The results showed that formulas agree well with the numerical results. In the high lift supply systems, compared with the other closure laws, the minimum speed and minimum pressure along the pipeline under valve closure formula were the largest, and the maximum pressure along the pipeline was the smallest. Moreover, in the low lift supply systems, the minimum speed under valve closure formula did not exceed the rated speed. Compared with the other closure laws, the minimum pressure along the pipeline was the largest and the maximum pressure along the pipeline was the smallest.


GRAPHICAL ABSTRACT INTRODUCTION
Long-distance pressurized water supply system is the most effective approach to solve uneven distribution of water resources (Wang et al. a, b; Ni et al. ). It is pressurized by pumps and then piped to transfer water resources from low-altitude areas to high-altitude areas. Therefore, cross-regional long-distance pressurized water supply systems can be seen all over the world (Sun et al. ; Miao et al. ). However, due to large flow and multi-objective water supply of the systems, the safety of pumps and along pipelines is a difficult problem. In order to regulate the flow and ensure the safety of pumps, valves are installed at the outlet of each pump. When pump trip happens, to prevent reverse speed of the pump from exceeding the standard, valves should be closed. Generally, the faster the valves are closed, the easier the reverse speed of the pump meets the requirements, but water hammer pressure caused by improper closure of valves can tremendously exceed pressure standard. If the pressure exceeds the pressure standard of the pipeline, it may cause enormous damage to the system (Adamkowski & Lewandowski ; Guo et al. ). Therefore, it is necessary to investigate how to effectively prevent the valves from being improperly closed of a long-distance pressurized water supply system. When pump trip happens in the long-distance pressurized water supply system, it is generally accepted that valves installed at the outlet of the pump are components to control the reverse speed of the pump by effectively regulating the flow (Wan et al. ; Essaidi &Triki ). Due to the fast and convenient regulating flow, the valves have been widely used in water supply systems all over the world (Bettaieb & Taieb ). In previous research, there are a large number of studies about the valve (Kim & Yoon ).
According to the closing rate of the valve, the valves are divided into slow closing valves and fast closing valves.
Slow closing valves include ball valves (Ferreira et al. ) and butterfly valves. The most common type of fast closing valve is the axial flow check valve (Yu & Yu ). There are many published papers regarding valve closure law (Li et al. ). According to the different curves of valve closure, the valve closure laws can be divided into one straight-line closure, two straight-line closure, and three straight-line closure. However, the above studies have focused on discussing the influence of valve closure curve on the water hammer pressure of long-distance water supply systems.
The long-distance water supply systems are mainly divided into pressurized and gravitational water supply systems. The gravitational system relies on topographical drop to supply water, and there is no pump in gravitational systems (Wang et al. a, b). Then, only the effect of different valve closure curves on the water hammer pressure of the system needs to be considered. However, compared with the gravitational system, the pressurized system relies on being pressurized by pumps to supply water. The pressure drop behind the pump is influenced by the characteristics of the pump. Hence, it is insufficient to only consider the effect of valve closure curves on water hammer pressure in long-distance pressurized water supply systems. Therefore, motivated by the above discussions, this paper focuses on the influence of valve closure law considering pump runaway characteristics on water hammer pressure in the long-distance pressurized water supply system.

MATHEMATICAL MODEL
For low Mach number fluids, the continuity and momentum equations of unsteady pipe liquid flow can be expressed as follows (Wylie et al. ; Chaudhry ): Equations (1) and (2) are converted to ordinary differential equations along the characteristic lines. The characteristic lines are shown in Figure 1. The characteristic equations can be obtained by integrating the ordinary differential equations.
The negative characteristic equation C À The positive characteristic equation C þ where, H Pi is unknown piezometric head of section i at time t. Q Pi is unknown discharge of section i at time t.

Valve model
The relationship between the head and discharge may be written as: A r is area of the valve opening (m 2 ). ΔH P is losses at the valve (m). (3) and (4), it can be obtained that:

Through Equations
As the Δt is small, the Q P at the right side of Equation (6) can be replaced by the instantaneous discharge rate Q P0 at the time t 0 ¼ t À Δt. Therefore, the discharge at the left can be directly obtained, and the values of H P1 and H P2 are determined.

Pump model
In long-distance pressurized water supply systems, pumps are important components. The Suter characteristic curves of a centrifugal pump are shown in Figure 2. WH(x) and WB(x) are separately the head and torque characteristics of the pump under different discharge and rotational speed conditions.  where, h ¼ H=H r , H is head of pump (m), and H r is the rated head of pump (m). n ¼ N=N r , N is speed of pump (r/min), and N r is the rated speed of pump. q ¼ Q p =Q p,r , Q p is the discharge of pump (m 3 /s), Q p,r is the rated discharge of pump (m 3 /s). m ¼ M=M r , M is the torque of pump (KN.m), M r is the rated torque of pump (kN.m).
The equation of speed for the pump where, J is moment of inertia (t·m 2 ), ω ¼ 2πN=60 ¼ nω r is angular velocity (rad/s), M g is the motor torque (KN.m).
For M ¼ mM g ¼ mP r =ω r , T a ¼ Jω 2 r =P r , P r is the rated output of pump (kW). Then, Equation (8) can be written as The integral of Equation (9) is obtained: Using the Taylor expansion with Equation (10): (1:5m 0 À 0:5m 00 ) where, m 00 is the relative value of torque at t 0 À Δt. m 0 is the relative value of torque at t 0 . m 00 and m 0 can be solved by Suter characteristic curves, respectively.
Therefore, the relative speed n at t can be obtained directly by Equation (11).
The head equation of the pump: where, H P1 , H P2 are piezometric head for the inlet and outlet of the pump, respectively (m).
According to Equation (7), it can be obtained that: Through Equations (3), (4), (12), and (13), we can get Relative discharge q can be calculated by Equation (14) using the Newton-Raphson method. According to q ¼ Q p =Q p,r , Q p can be obtained. Then, according to Equations (3) and (4), H P1 and H P2 can be obtained.

VALVE CLOSURE BASED ON PUMP RUNAWAY CHARACTERISTICS Theoretical analysis of pump runaway characteristic
A long-distance pressurized water supply system is shown in where, subscripts 0, 1, and 2 indicate the time after pump failures, which are time at time t ¼ 0, t ¼ L=a, t ¼ 2L=a, respectively.
By simplifying Equation (15), we can get Figure 3 | A long-distance pressurized water supply system layout.
Generally, the water level of the upstream reservoir can be constant to H U . Therefore, it is approximated that the change of pump head is equal to the change of piezometric head behind the pump. According to the above analyses, it can be obtained that h ¼ H=H r ¼ H 0 À ΔH=H r and By substituting h and q into Equation (7), the following expression can be obtained: When pump trip happens, the pump is runaway at time t ¼ 2L=a. According to Equation (7) By combining Equations (16) and (17), we can get where, i is the number of the same parallel pumps Based on the above analyses, when pump trip happens, the pump is in the runaway condition at time t ¼ 2L=a, and the pump must be at point a or point b.
If the pump is at point a, then x ¼ x a . The pump has reverse speed and reverse flow.
has positive speed and positive flow at point b.
In summary, when the long-distance pressurized water supply system is high lift, then H B À H U À aQ 0 =gA > 0, the pump must be at point a at time t ¼ 2L=a, and the pump has reverse speed and reverse flow. On the contrary, when the long-distance pressurized water supply system is low lifted, pump is influenced by the characteristics of pump and the valve closing law. If the closing time of the valve is T g , and T g < 2L=a, then, direct pump-stopping water hammer behind the valve can be obtained.
where, ΔV s ¼ (V 0 À V 2 ), V 2 is the flow velocity throughout the pipeline at time t ¼ 2L=a (m/s).
According to the analyses, when pump trip for the system with H B À H U À aQ 0 =gA > 0 happens, the four quadrant of the zones of pump operation is the reversed speed dissipation at time t ¼ 2L=a. And the pump has reverse speed and reverse flow. Thus, V 2 < 0 and ΔV s > ΔV. If T g < T d , according to Equations (19) and (20), it can be obtained that ΔH P ¼ ΔH < ΔH s . ΔH P is maximum value of the pressure drop behind the pump under the valve closing.
Based on the above analyses, it can be believed that, when H B À H U À aQ 0 =gA > 0, fast closure of valves is beneficial to the safety of long-distance pressurized water supply systems.
When pump trip for the system with H B À H U À aQ 0 =gA < 0 occurs, the pump has positive speed and positive flow at time t ¼ 2L=a. Thus, V 2 > 0 and ΔV s < ΔV. If T g > T s , according to Equations (19) and (20), it can be obtained that ΔH P ¼ ΔH s < ΔH. Based on the above analyses, it can be believed that, when H B À H U À aQ 0 =gA < 0, slow closure of valves is beneficial to the safety of long-distance pressurized water supply systems.
Based on the above analyses and discussions, the valve closure formulas considering pump runaway characteristics can be obtained.
where, f(t) is the valve closure formula.

CASE STUDY AND ANALYSIS
The pump runaway characteristics are theoretically analyzed in long-distance pressurized water supply systems.
According to the theoretical analyses, valve closure formulas considering pump runaway characteristics are proposed. In this section, the formulas are verified using numerical simulation based on practical projects. Layouts of the long-distance pressurized water supply systems are shown in Figure 5.  Table 1.

Verification of theoretical analyses
The elevation and piezometric head along the pipeline of Case 1 are shown in Figure 6(a). In order to verify the accuracy and rationality of the valve closure, Equation (21)  According to Table 1, the initial flow velocity is 0.974 m/s. (19), it can be obtained that shows that the theoretical analyses are reasonable.

Sensitivity analyses
According to Equation (21)     valves less than T d and greater than T d , these results show that, when T g ¼ T d ¼ 1:0 s, the minimum speed is the largest, and the minimum pressure along the pipeline is the largest, and the maximum pressure is the smallest. Based on the above analyses, it can be believed that, for the system with H B À H U À aQ 0 =gA > 0, both the pump and Valve closure formula for low lift supply systems

Verification of theoretical analysis
The elevation and piezometric head along the pipeline of Case 2 are shown in Figure 5(b). In order to verify the valve closure Equation (22)  According to Table 1, the initial flow velocity is 1.03 m/s. based on Equation (19), it can be obtained that believed that, for the system of Case 2, which is

Sensitivity analysis
According to Equation (22), the closing time of the valve should be longer than T s . From Figure 8(a), T g > T s ¼ 36 s.
In order to verify the valve closure Equation (22), the sensitivity of valve closing time T g is analyzed. The analysis schemes are shown in Table 3. Under these analysis schemes, variations of pressure behind the pump, discharge of pump, speed of pump, maximum and minimum pressure curves along the pipeline are shown in Figure 9.

DISCUSSION
In this paper, in order to reduce the water hammer pressure caused by pump trip in long-distance pressurized water supply system, the valve closure formulas considering pump runaway characteristics are derived. Then, the formulas are verified using numerical simulation based on practical projects. In this section, the effects of valve closure formulas on water hammer pressure are discussed.
1. The effects of the characteristics of long-distance pressurized water supply system on the pump runaway are    sure laws, the minimum speed is À1,241.77 r/min, which does not exceed the rated speed. Moreover, minimum pressure along the pipeline is the largest, which is À28.0 m, and the maximum pressure along the pipeline is the smallest, which is 136.8 m. Therefore, it can be believed that, for the low lift system, the pump is safe, and the pressure along the pipeline is the safest under the valve closing Equation (22).