Research on the Flow Resistance Coecient of a Multi-Hole, Secondary Pressure-Reducing Sleeve Valve

To solve the problem of valve noise, a multi-hole sleeve valve with secondary pressure-reducing function is presented in this paper. During the ow design of the valve, the ow resistance coecient of the valve served as an important parameter. Because of two pressure-reducing components assembled to a multi-hole sleeve valve, the ow resistance coecient of the valve changed. Thus, correction of the ow resistance coecient had to be affected. In this paper, the relationship between the ow rate and ow resistance coecient of the valve was rst mapped and established. Then, the ow rate of the sleeve was obtained using SolidWorks simulation software. Locally rened nite element mesh technology was applied to the simulation to improve simulation accuracy. A parallel ow test platform for the regulating valve was established, and the ow rate of the multi-hole sleeve valve was detected at different openings, thus, verifying the reliability of the numerical simulation results. Finally, the simulation ow rate of the valve at different openings was substituted into the mapping relationship formula, in this way, the ow resistance coecient of the sleeve valve was obtained. By using the modied ow resistance coecient, the ow rate characteristics of the multi-hole, secondary pressure-reducing sleeve valve were eciently and accurately established.


Introduction
A regulating valve is a control composite that assists in controlling the ow rate, throttling, and stabilizing pressure in a control system [1][2][3] . In recent years, alongside the development of science and technology, the requirements for the performance of regulating valves have also continuously improved.
Environmental protection requirements, particularly reducing noise pollution of the regulating system, have also been put forward. When uid ows through the throttle hole of the regulating valve, as the ow area decreases, the ow velocity increases, and the pressure difference between the two sides of the throttle holes rises. This can cause signi cant noise and effect damage to control equipment [4][5] .
The sleeve valve [6][7] is a specially structured regulating valve. The valve plug moves up and down in a cylindrical sleeve to change the ow area, thereby controlling the ow rate of the valve. To control and reduce pressure difference and noise, the throttle holes of sleeve valves have been designed as the labyrinth [8][9] , window [10] and multi-hole types [11] . However, the labyrinth throttle hole type signi cantly restricts ow rate. The window-type sleeve valve has a poor noise reduction ability. In this paper, a multihole sleeve valve with a secondary pressure-reducing function is presented. A set of pressure-reducing components were assembled inside and external to the throttle holes, respectively, so that it had low noise, good dynamic stability, and other advantages. However, when we designed the ow characteristics of the valve, the ow channel and the pressure-reduction components had a signi cant impact on the ow resistance coe cient. A large error occurred when we employed a traditional ow resistance coe cient to calculate the ow rate at different openings. However, calculating using the nite element method, or simulation method, can be very complex. Additionally, each simulation required signi cant time to complete and hardware with powerful computational capabilities. As such, it was necessary to establish an e cient theoretical method to design the ow rate of the multi-hole, secondary pressurereducing sleeve valve. In the theoretical calculation process, the ow resistance coe cient of the valve directly affected the calculation accuracy. Accordingly, in this paper, the ow rate of the valve at each opening was obtained by numerical simulation. The simulation results were veri ed using a ow rate test. Then, the simulation ow rate was substituted into the mapping relationship equation between the ow rate and the ow resistance coe cient of the valve. In this manner, the ow resistance coe cient of the sleeve valve was obtained.
The modi ed ow resistance coe cient can be used to design this type of valve with different ow and diameter characteristics.
2 Structural Description Of The Multi-hole, Secondary Pressurereducing Sleeve Valve The structure of the multi-hole secondary pressure-reducing sleeve valve is shown in Fig. 1.
It included eight major components: the valve body, the valve seat with noise reduction cage, external noise reduction cage, multi-hole sleeve, pressure cage, valve plug, valve rod, and a bonnet. The outer surface of the valve plug and the inner surface of the multi-hole sleeve represented tted surfaces. Under the action of the external actuator, the valve rod was able to drive the valve plug to move up and down, in this manner, the throttle holes on the multi-hole sleeve were exposed to form an effective uid area. The size and layout of these holes were able to realize the different ow characteristics of the regulating valve. When the pressure difference between the two sides of the throttle holes was large, ash evaporation and cavitation could occur [12][13] , giving rise to signi cant noise and vibration. To reduce the valve noise, a valve seat with a noise reduction cage and an external noise reduction cage was assembled on both sides of the multi-hole sleeve.

Flow Calculation Of The Multi-hole, Secondary Pressure-reducing Sleeve Valve
The ow rate through the throttle holes was calculated according to hydromechanics. Based on the thickness of the sleeve, the throttle holes on the multi-hole sleeve were typically thin-walled. These thinwalled holes were short with minimal frictional resistance. The ow rate was minimally affected by temperature and viscosity changes and, as such, was relatively stable.
The ideal uid passed through a thin-walled hole, as shown in Figure 2.
First, we assumed that uid energy loss had been ignored when the uid passed through the ow channel. According to the law of the conservation of energy, the total energy of the uid at the inlet of the valve was equal to the total energy of the uid at the outlet. In the cross-sections and , according to the Bernoulli equation, the energy of the uid can be expressed as follows: 1 where the pressure difference of the uid in cross-sections and : Δp = p 1 -p 2 ; p 1 , p 2 is uid pressure in the cross-sections and , respectively; h 1 , h 2 is the potential energy of the uid in cross-sections and , respectively; v 1 , v 2 is the ow velocity of the uid passing through cross-sections and , respectively; ρ is the density of uid; ξ is the ow resistance coe cient; α 1 , α 2 is the kinetic energy correction coe cient in cross-sections and , respectively; The diameter of cross-sections and is d; the diameter of the contraction section of the thin-walled hole is d 0 . Owing to d » d 0 , thus, v 1 ≈ 0. When the uid owed through the contraction section of the thinwalled hole, the ow velocity was uniform, thus, α 1 = α 2 = 1, and the uid potential energy in crosssections and was equal, i.e., h 1 = h 2 .
According to Eq. (1), the ow velocity of the uid owing through cross-section was obtained as follows: 2 The sectional area of the thin-walled hole is A e , and the sectional area of cross-sections and is A.
According to Eq. (2), the ow rate of the uid passing through the hole was obtained as follows: Equation (3) shows the ow rate of the uid through the sleeve valve was closely related to density ρ of the uid, effective ow area A e , pressure difference Δp, and ow resistance coe cient ξ.
When the uid owed through the valve seat with the noise reduction cage, the external noise reduction cage, and the throttle hole of the multi-hole sleeve, their combined multi-hole structure caused part of the energy to be lost and reduced uid pressure, changing the ow resistance coe cient. Accordingly, a large error will be observed when using traditional ow resistance coe cient ξ to calculate the ow rate of the valve, which must be corrected. Same-stepped holes were equally distributed around the valve seat and were also equally distributed on the external noise reduction cage. These small holes were prevent ash evaporation and cavitation, and could effectively reduce the pressure and noise caused by the uid. The ow area of these holes had be 5% larger than that of the throttle holes when the sleeve valve was fully open. This enabled them to reduce pressure and noise caused by the uid to a satisfactory degree, without affecting the ow rate of the sleeve valve.

Finite element mesh
The SolidWorks ow Simulation module was used to set the global nite element mesh to the highest level, as shown in Fig. 3.

Boundary conditions and simulation settings
Based on the de nition of ow capacity Kv of the regulating valve [14][15][16] , the simulation boundary conditions were set. When the regulating valve was fully opened, pressure difference Δp at the inlet and outlet of the valve was 100 KPa, and the uid density was 1000 kg/m 3 (room temperature water); ow capacity Kv was the ow rate of uid that passed through the valve in 1 h. The pressure at the inlet of the valve was set to 201,325 Pa, the pressure at the outlet was 101,325 Pa, and the roughness of the inside surface of the valve body was Ra25. The insertion target was the volume ow rate through the crosssection of the valve outlet.

The ow rate simulation analysis
The above model was used to simulate the multi-hole, secondary pressure-reducing sleeve valve. Its ow capacity was Kv = 38.06. The maximum stroke of the valve plug was 38 mm. The model of the valve that conformed to linear ow characteristics was tested in the simulation. 4.4.1 Static pressure distribution of the uid inside the valve. When the valve was fully opened, the static pressure distribution cloud diagram of the internal uid on the symmetrical section was obtained, as shown in Fig. 4. 4.4.2 Flow rate simulation of the valve. The volume ow at the outlet of the valve was monitored. Following the iterative calculation, the volume ow rate at an opening ranging from 10-100% was obtained.
The simulation ow rate was compared with the theoretical standard ow rate, which met the standard linear ow characteristics. The curve subsequently obtained is shown in Fig. 5.
The maximum error occurred at an opening of 10%, the error value was 5.55%, and the minimum error occurred at an opening of 100%. At 90% and 100% openings, the ow rate of the valve was lower than the standard value, the ow of other openings was slightly larger than the standard value. The software simulation method was used to effectively verify whether the ow rate of the valve met the speci ed ow characteristics. However, following the mesh re nement, the number of nite element meshes increased sharply, and each calculation required an extended period to nish iteration prior to achieving convergence. Concurrently, hardware support with powerful computing capabilities was required. As such, using software simulations to design the valve ow rate presented some limitations. To test whether the regulating valve met the rated ow characteristics [17][18] , a parallel ow rate test system with recyclable uid equipment was designed, as shown in Fig. 6. The system comprised a water storage tank, parallel multi-stage pump, a surge tank, parallel test area, and a backwater pipe. The parallel test area comprised multiple test pipelines connected in parallel, which were able to test the regulating valves with nominal diameter, ranging from DN15 to DN450. When a pipeline was active, the manual ball valves (3) at both ends of the remaining pipelines were closed. During the test, the regulating valve was connected to the position indicated by 4 in Fig. 6. The pressure gauges (5) were set at the inlet and outlet of the tested regulating valve, and the data were transmitted to the computer. The computer controlled the pipeline pump (10) through the inverter to adjust the water pressure. If a greater pressure was required, several pipeline pumps could function simultaneously in parallel, as shown in Fig. 7. The electric pressure control valve (2) was set at both sides of the tested regulating valve to regulate the pressure at the inlet and outlet of the valve.
When the pressure difference met the requirements, the ow rate at different openings of the tested valve could be read from the electromagnetic owmeter. The uid of each test line was sent back to the water storage tank through the return pipe.

Test data analysis
The tested multi-hole, secondary pressure-reducing sleeve valve had the same parameters as the simulation model. The pressure difference between the inlet and outlet of the valve was 100 KPa. The ow rate of each opening from 10-100% was observed from the electromagnetic owmeter, as shown in Table 1. According to Eq. (3), the following equation was obtained: 4 where ξ i is the ow resistance coe cient of each opening; q vi is the ow rate of each opening; A ei is the ow area of the throttle holes of each opening.
i is the opening, i.e., i = 1 to 10.
The pressure difference at the inlet and outlet of the valve can be given as Δp = 100 KPa, and the uid density as ρ = 1000 kg/m 3 (room temperature water). According to Eq. (4), q vi , A ei , and ξ i of the multi-hole, secondary pressure-reducing sleeve valve at different openings could be calculated, as shown in Table 2.  Table 2 indicates that at openings of 20-100%, the ow resistance coe cient value ξ of the multi-hole, secondary pressure-reducing sleeve valve was roughly 2.2. At a 10% opening, the value of the ow resistance coe cient ξ was 3.78. Both the simulation results and test data re ected that the ow resistance coe cient was larger and the ow rate smaller. Therefore, based on current study, the value of the ow resistance coe cient ξ of the multi-hole, secondary pressure-reducing sleeve valve was determined as a constant value, i.e., 2.2. When we designed the ow rate of the valve, to avoid the ow rate at a 10% opening from being insu cient, the ow resistance coe cient could be increased 1.5 times, that is, this could be achieved by increasing the ow area. According to the above research, as long as the effective uid area of the throttle holes was known, a multi-hole, secondary pressure-reducing sleeve valve conforming to different ow characteristics and different C v values could be obtained.
(1) It is concluded that the ow resistance coe cient of the valve was 3.3 at a 10% opening, and 2.2 at an opening ranging 20-100%. By using the modi ed ow resistance coe cient, the multi-hole, secondary pressure-reducing sleeve valve could be designed more e ciently.
(2) Using the mesh re nement technology of the local space virtual model, the simulation of the ow rate of the multi-hole, secondary pressure-reducing sleeve valve was performed. Experiments veri ed that the error of the simulation results was within the allowable range. The simulation ow rates were also concluded as being reliable.
(3) A parallel ow rate test system with recyclable uid equipment was designed. The system was able to test the ow rate of the regulating valves with different speci cations. The ideal differential pressure data could be adjusted quickly during the test, and the test results were subsequently more reliable.

Declarations
Availability of data and materials All data, models are available from the corresponding author by request.   Simulation model of the multi-hole, secondary pressure-reducing sleeve valve 1-the valve body, 2-the valve seat with noise reduction cage, 3-external noise reduction cage, 4-multi-hole sleeve 5-pressure cage, 6-valve plug, 7-valve rod, 8-bonnet.

Figure 4
Finite element mesh of the model Figure 5 Cloud diagram of static pressure distribution Figure 6 Comparison of simulated ow rate with standard data Figure 7 Regulating valve ow test system 1. backwater pipe; 2. electric pressure control valve; 3. manual ball valve; 4. test point of regulating valve; 5. pressure gauge; 6.electromagnetic owmeter; 7. diverging pipeline; 8. surge tank; 9. expansion joint; 10. pipeline pump; 11. diverging pipeline;12. water storage tank Pipeline pumps in parallel