An Accurate Mathematical Model and Experimental Research of Pressure Distribution in the Spool Valve Clearance Film

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Introduction
As the amplifer stage of an electrohydraulic system, the spool valve provides control over the direction and fow of the fuid, which is widely used in various hydraulic actuators that can be utilized to control the aircraft rudder surface defexion, control the drilling steering, control the turning of a ship, and so on. Terefore, the spool valve afects the hydraulic actuator's position control accuracy, speed control accuracy, and force control accuracy. However, the spooltype valves are susceptible to a unique problem known as hydraulic lock due to the uneven pressure distribution in the clearance flm between the spool and the sleeve. Te uneven pressure distribution around the spool creates a lateral force which causes the spool to tilt at a small angle. Ten, the lateral force increases with the tiny angle and makes the tilted angle greater so that the spool contacts with the inner wall of the sleeve, resulting in the spool locking in the sleeve. Te spool locking phenomena cause the actuator of hydraulic equipment to fail and even cause safety accidents, as shown in Figure 1. In theory, the locking problem could be avoided when the clearance exactly parallels with the sleeve's walls. However, in practice, the perfectly parallel walls between spools and sleeves are not obtainable due to the limitations of tooling machines. Fortunately, some circumferential grooves applied on the spool surfaces can be used to decrease the lateral force to limit the efects of locking phenomena.
Te frst study about limiting the hydraulic locking phenomena was conducted by Sweeney [1], who presented that some grooves along the spool axial direction can mitigate uneven pressure distributions in the clearance flm. Since then, several researchers [2][3][4][5][6][7][8][9][10] have utilized the Reynolds equation to investigate the lubricant characteristics of the clearance flm of the spool valve. For instance, the pressure in each groove and the leakage in the clearance flm were calculated by Milani [10] based on the Reynolds equation. Since the Reynolds equation is derived from the NS equation, Dong et al. [11] compared the diference of pressure distribution between the Reynolds equation and the NS equation and found that the diference exceeds 20% when the clearance flm is greater than 0.413 μm. Furthermore, Guardino et al. [12] investigated the diference in various values of the roughness amplitude to the clearance flm ratio and pointed out that the diference increases with the ratio. In the following years, some research results [13][14][15][16][17][18] used in hydrostatic gas-lubricated tribological components obtained the same results as those in reference [12]. Simultaneously, the research methods based on the NS equation are applicable to the study of the clearance flm lubrication between the sleeve and the spool as the depth of the groove to the thickness of the clearance flm ratio is greater than ten [19,20]. Terefore, Hong and Kim [20] used computational fuid dynamics (CFD) based on the NS equation to compare spiral grooves' migrating uneven pressure distribution efects and traditional grooves. Te uneven pressure distribution was simulated for diferent inlet pressure conditions [21]. It was found that the uneven pressure increased linearly with the inlet pressure. Unfortunately, the study [21] did not investigate the efect of grooves on eliminating uneven pressure. Meanwhile, CFD was widely utilized to study the efect of diferent types of grooves on improving load-carrying capacity and flm stifness in gas face seals [22][23][24][25]. Load-carrying capacity and flm stifness in the gas face seal are related to the type of grooves and the geometric values of grooves. However, loadcarrying capacity and leakage rate are conficted. In other words, the improvement of load-carrying capacity increases the leakage rate, which must be as small as possible for the gas face seal. Aiming to take into account load-carrying capacity and volume fow leakage, Wang et al. [26] frstly established an accurate mathematical model of grooves and then utilized this mathematical model to optimize the groove shapes using the multiobjective optimization approach, which is very mature in the present time. Tus, the accurate mathematical model of grooves is the most important for the multiobjective optimization of groove shapes. Unfortunately, the current mathematical models of the clearance flm pressure distribution between the sleeve and the spool are based on the Reynolds equation. Te diference of pressure distribution in the spool valve between the Reynolds equation and the NS equation increases with width, depth, and number of grooves [27].
Terefore, an accurate mathematical model of clearance flm pressure distribution in the spool valve is established based on the NS equation in cylindrical coordinates in this paper. Meanwhile, this accurate mathematical model takes into account the distribution, width, depth, and the number of grooves and is used to investigate the lateral force and the volume fow leakage under various groove geometric dimensions. To verify the mathematical model, an experiment is tested to compare with the theoretical results obtained by the NS equation and the Reynolds equation, respectively.

Description of the Spool Valve Structure
Te spool valves are utilized to amplify the hydraulic energy in the servo valve, as shown in Figure 2(a), and to control the fow's direction and fow in the spool valve system, as shown in Figure 2(b). Figure 2(c) illustrates that the spool valve consists of the sleeve and the spool, and a spool usually has several lands. For the convenience of calculation, the mathematical model of one land is frst established. Tus, the geometry of the spool can be simplifed as Figure 2(d). Te simplifed spool with radius r sp and length 2l is nested in the sleeve with the radius r sl . Te spool is stationary with a tilted angle α in the sleeve.
Te classical groove applied to the spool mostly forms a circular groove with a rectangular cross-sectional area. Te parameters of the groove consist of the distance between the groove and the spool edge, the distance l 3 between grooves, the width l 2 of groove, and the depth h z of the groove, as shown in Figure 2(d). Supposing that the number of the grooves is n, the length of the spool can be written as Te geometric dimensions of the spool and the groove are tabulated in Table 1. Te length of spool l, the radius of the spool r sp , the inner radius of the sleeve r sl , and l 1 are all constant, which do not afect the lubricant characteristics of the clearance flm between the spool and the sleeve. Besides, the aspect ratio K represents the groove depth h z to the groove width l 2 ratio, the width of the spool l 2 , the distance between grooves l 3 , and the tilted angle α which are variable geometric dimensions of the spool, which need to investigate the efect on the lubricant characteristics.

Mathematical Model
As shown in Figure 2(d), taking into account the cylindrical shape of the spool and the sleeve, the NS equations in cylindrical coordinates are written as  Mathematical Problems in Engineering Te sleeve and the spool are both assumed to be stationary. In addition, the tilted angle of the spool is small enough that the fow in the clearance flm between the spool and sleeve moves approximately along the z-direction. Te terms for velocity in the NS equation thereby can be written as When (4) is substituted into (2), and the second-order diferential terms in (2) are reduced, (2) can be written as Te integral general solution of fuid velocity v z can be listed as follows: where μ is the fuid's dynamic viscosity, r is the radius with O as the center, and C 1 and C 2 are both constants. According to the assumption above, v z equals zero when r in (6) equals the radius r sl . When r equals r z which is the distance from the center O to the wall of the spool, r z equals zero since the spool is stationary. r z is a variable parameter, the value of which is related to the center distance OO sp and the rotary angle θ, that needs to be calculated using the longitudinal section illustrated in Figure 3. Due to the very small tilted angle α, the longitudinal section of the spool is approximately circular whose center is O sp , and the radius is r sp ′ � r sp cos α. As shown in Figure 3, the eccentricity direction is assumed to coincide with the angles increasing in the clockwise direction. Ten, the analytical expression for r z can be deduced as follows: Introducing the boundary conditions and putting (7) into (6) by linearizing ln r to ζr, the fuid velocity v z in the clearance flm becomes where the term ζr is eliminated in the calculation, and thus, the efect on the calculation of linearization above is ignored. Te diferential element of volume fow leakage in the clearance flm can be written as where the range of r is from r z to r sl . Integrating (9), Q can be expressed as  Introducing (8) into (10), the integration for Q is solved as follows: According to (11), the analytical expression of the pressure gradient along the z-axis becomes dP dz � 8μQ Since the range of spools in the z-axis is from −l to l, the pressure distribution in the clearance flm can be written in (13) by solving a defnite integral pressure gradient: where C 3 is a constant, and f(z) is a function with z as its argument. According to an integral part of (12), f(z) is expressed as For the pressure distribution at both ends of the spool, the frst boundary conditions are P � P 2 and z � l, and the second boundary conditions are P � P 1 and z � −l. Tus, owing to the continuity of fuid, the constant C 3 can be written as Te volume fow leakage in the clearance flm is written as Tus, from equation (13) to (15), we can obtain the following: Te value of h z in (14) is related to the z-axis coordinate value and the number of grooves. When the number is even, the value of h z becomes where m is the number of grooves, and the value can be set as 1, 2, 3, . . ., n/2.

When the number is odd, the value of h z becomes
where the value of m can be set as 1, 2, 3, . . ., (n + 1)/2, Equation (17) presents the pressure distribution in the clearance flm between the spool and the sleeve, which can be used to calculate the lateral force by integrating the pressure P in the clearance flm, as shown in (20). Te decrease rate of the absolute value of the lateral force refects the efect of eliminating the uneven pressure distribution around the spool [21]. Te smaller the lateral force, the better the efect of eliminating the uneven pressure distribution:

Te Optimal Distribution of the Grooves.
Te optimal distribution of the grooves needs to be confrmed frst. For the sake of investigating the efect of parameter l 3 on the pressure distribution in the clearance flm, several cases are compared through calculating the pressure distribution and the lateral force based on the geometric dimensions tabulated in Table 1. Except for the parameter l 3 , each case utilizes the same geometric dimensions, such as width (l 2 � 0.8 mm), depth (K � 0.25), and the number of grooves. Meanwhile, each case shares the same properties of the working fuid, as shown in Table 2, and utilizes diferent pressure conditions, as shown in Table 3. With the decrease of l 3 , the grooves are clustered to the end edge of the spool (l 3 � 0.1 mm). When l 3 equals 1.6 mm, the grooves are evenly distributed. Figure 4(b) illustrates that the lateral force has a minimum value in the case where the grooves are evenly distributed. And the equidistribution grooves have the same efect on the pressure distribution in the case of diferent pressure conditions and tilted angles, as shown in Figures 4(c) and 4(d).
Figures 4(c) and 4(d) also illustrate that the decreased slope of the lateral force with an increment of l 3 under a tilted angle, α � 0.0188°, is lower than that under the tilted angle α � 0.0288°. Tis is because the efect of grooves decreases with the reduction of tilted angle α. Nevertheless, compared with the lateral force at l 3 � 0.1 mm, the lateral force at l 3 � 1.6 mm is reduced by 16.9%, 14.9%, and 14.8% at pressure 5 MPa, 7 MPa, and 9 MPa, respectively. Same as the value of the lateral force, the uniformly distributed grooves provide a more efective lubricant characterized along the circumference, mitigating the unbalance pressure distribution. Figure 4(a) supports the efect, which is very approximate to the numerical results in reference [27]. Terefore, compared with other distributions of grooves, the even distribution groove is optimal.

Te Efect of the Cross-Sectional of Groove on Lateral Force.
Besides the distribution of grooves, the geometric dimensions also afect the value of the lateral force, such as the groove depth h z and the groove width l 2 [27]. Terefore, it is essential to investigate the efect of these two parameters on the lateral force. As shown in Figures 5-7, the lateral force decreases with the increase in the width of the groove for the same value of K in cases where the number of grooves is 2, 4, 6, and 8, respectively. Meanwhile, the lateral force also decreases with the increment of the aspect ratio K for the same value of l 2 . Tis is because the lateral force is related to the cross-sectional area which is proportional to the width l 2 and K, as shown in Figure 8. In other words, the increase in the value of the cross-sectional area indicates a decrease in the lateral force value. As a result, the increase in crosssectional helps migrate the uneven pressure distribution surrounding the spool.
In addition, diferent increasing rates of cross-sectional area of the groove lead to diferent decreased slopes of the lateral force. For instance, as shown in Figure 5(a), the greater value of K leads to the greater increase rate of the cross-sectional area, which causes the greater decreased slope of the lateral force for the same increased rate of l 2 . Similarly, diferent groove numbers result in diferent decreased slopes of the lateral force. As shown in Figure 9, for the same value of K and the same increased rate of the groove width l 2 , the decreased slope of the lateral force increases with the increase in the groove number in the case where the pressure P 1 � 7 MPa and the tilted angle α � 0.288°. For example, in the case where K � 0.25, along with the same increased rate of l 2 , the reduction rate of the lateral force is 3.35%, 5.91%, and 17.49% when the number of the grooves is eight compared with which is 1.73%, 2.89%, and 4.4% when the number of the groove is 4. Tis conclusion also applies to the reduction rates of the lateral force in cases of P 1 � 5 MPa and 9 MPa, respectively, and the tilted angle α � 0.188°, as tabulated in Tables 4-6.
According to the theoretical results above, the lateral force decreases with the increase in the groove number. However, the volume fow leakage increases with the increase in the groove number, which is illustrated in Figure 10. Meanwhile, the volume fow leakage increases with the increase of P 1 since the volume fow leakage is positively proportional to the diferential pressure according to (16). As the sole driving energy of the spool, the diferential pressure, as shown in (21), decreases when the volume fow leakage is too much. In other words, too much volume fow leakage causes the reduction of the driving force of the spool, to reduce the dynamic characteristic of the valve: In addition, the strength of the spool reduces with the increase in groove depth. Terefore, the width of the groove, the depth of the groove, the number of the grooves, the leakage volume fow, and the strength of the spool need to be comprehensively considered when optimizing the parameters of the groove.

Experimental Verification
As for the second or third stage of the servo valve or other valves, the spool usually has more than one land. To conclude more closely to ft the actual situation, the second stage spool of the servo valve with four lands is investigated. Figure 11 shows the spool valve, and the spools with four lands are named as S 1 , S 2 , S 3 , and S 4 . As shown in Figure 11, P 1 and P 2 are the relatively high-pressure inlet and the  relatively low-pressure outlet, respectively, and P A and P B are the loading outlet of the servo valve. Te hydraulic diagram of the test bench is shown in Figure 12. Te two overfow valves are used to change the supply oil pressure. Te solenoid valve 4 is normally open. When the solenoid valves 1 and 2 are both turned on, and the solenoid valve 3 is turned of, the volume fow leakage between the land S 2 and the sleeve can be tested by fowmeter. Conversely, when the solenoid valves 1 and 3 are both turned on, and the solenoid valve 2 is turned of, the volume fow leakage between the land S 3 and the sleeve can be tested by fowmeter.
To verify the mathematical model derived above, the volume fow leakage rate between the spool and the sleeve is tested by the equipment setup illustrated in Figure 13. Te test system consists of the oil supply system, the spool valve drive system, and the data acquisition system. In Figure 13(b), the spool valve drive system is composed of the stepper motor, and the spool valve is preinstalled and tilted at an angle α. Te stepper motor propels the spool towards the positive direction of the z-axis until the port A is fully open. Meanwhile, port A is connected to the relatively high port, and port B is connected to the relatively low-pressure port. Ten, the volume fow leakage between the land S 2 and the sleeve is measured by the fowmeter, that is shown in Figure 13(c). Conversely, when the spool was pulled in the negative direction of the z-axis, the volume fow leakage rate between the land S 3 and the sleeve was measured and recorded. Figure 13(d) presents the oil circuit connection photo at the bottom of the spool valve. Tus, S 2 , S 3 , and the sleeve can be obtained at diferent diferential pressure values derived from the impairment between P 1 and P 2 . Te oil supply system can provide diferent constant values of pressure, where the value of return oil is 0.3 MPa so that the value of P 2 is set as 0.3 MPa in the mathematical model. Te boundary conditions of the land S 2 are shown in     Figure 14. To keep the boundary conditions consistent with the theoretical calculation, the relatively high-pressure P 1 in the experiment is set as 5.3 MPa, 7.3 MPa, and 9.3 MPa, respectively, taking the spool valve used in hydraulic dives and control, where working pressure often reaches 35 MPa. Te relatively high-pressure P 1 which equals 35.3 MPa is considered both in the simulation and experiment. Ten, the diferential pressure P d in the experiment is the same as that in the theoretical calculation. Simultaneously, the number of the groove and the tilted angles of the spool are both taken the same values as the values of the theoretical calculation, as tabulated in Table 1.    According to the boundary conditions above, the volume fow leakage rate of the theoretical results and experimental results is compared in Figure 14. Te theoretical results and the experimental results both present that the volume fow leakage rate increases with the increase in the number of grooves and the diferential pressure. In addition, since under the same inner diameter of the sleeve, the greater the tilted angle of the spool, the smaller the clearance flm between the spool and the sleeve, the volume fow leakage rate of the theoretical results decreases with the increase in titled angle, which is just same as that of the experimental results.
Meanwhile, Figure 15 shows that the volume fow leakage of the theoretical results is smaller than that of the experimental results in the cases where the diferential pressure is 5 MPa and 7 MPa, but it is the opposite in the case where the diferential pressure is 9 MPa and 35 MPa, respectively. With the increase of the diferential pressure, the linearity of the theoretical volume fow leakage value is better than that of the experiment, which is more clearly shown in Figure 15. Tis is because the dynamic viscosity of the fuid increases with the increase in diferential pressure, which is ignored in the theoretical model. Nevertheless, the theoretical results are close to the experimental results, which verify the mathematical model derived above.
As shown in Figure 15, the "NS equation" and "Reynolds equation" represent the results obtained by the NS equation and the Reynolds equation from reference [21], respectively. Te comparison results show that the volume fow leakage obtained by the NS equation is closer to the experimental results than that obtained by the Reynolds equation. For instance, the maximum percentage of the diference between the results obtained from the Reynolds equation and experimental results is 14.75% in the case where the diferential pressure P d is 5 MPa, the groove number n is 6, and the tilted angle α is 0.0188°. Under the same condition, the percentage of the diference between the results obtained from the NS equation and experimental results is 5.57%. Tat is to say, the mathematical model of the spool valve clearance flm proposed in this paper is more accurate than the models derived from the Reynolds equation.

Conclusions
Tis study establishes an accurate mathematical model of the pressure distribution in the clearance flm between the sleeve and the spool with rectangular grooves. Tis model was derived from the NS equation in cylindrical coordinates and can be used to calculate the uneven pressure distribution, lateral force, and volume fow leakage for the incompressible, isothermal, and Newtonian fow in the clearance flm between the sleeve and the spool in diferent confgurations such as geometric dimensions of the spool width, depth, and number. In addition, the values of the volume fow leakage obtained by the mathematical model are compared with that obtained by the experiment under various boundary conditions. Te theoretical results and the experimental results indicate conclusions as follows: (1) Compared with the experimental results, the mathematical model derived from the NS equation could more accurately calculate the lubricant characteristics of the fuid in the clearance flm between the sleeve and the spool with rectangular grooves than the model derived from the Reynolds equation in the case of various parameters. However, the efect of the dynamic viscosity changes due to the diferential pressure on the uneven pressure distribution required in the future. (2) Te uniformly distributed grooves provide a more efective lubricant characterized along the circumference, mitigating the unbalance pressure distribution. Furthermore, the theoretical results suggested that the wider, deeper, and more number of grooves could more efectively mitigate the unbalanced pressure distribution. In other words, an increase in the cross-sectional area of the groove could efectively decrease the lateral force acting around the spool.
Although the increase in cross-sectional area of the groove could reduce the lateral force, it also degrades the performance of the spool valve. For instance, the increase in volume fow leakage owing to the increment of the number of grooves reduces the driving energy of the spool. Meanwhile, the increment of the depth of the groove reduces the structural strength of the spool. In addition, the mathematical model established in this paper neglects the variation of dynamic viscosity with pressure although the theoretical results are close to the experimental results.

Data Availability
Te (groove_with_L2_change_n_8.m) data used to support the fndings of this study are included within the article. Te (groove_with_L2_change_n_8.m) data present the lateral force in the case where P1 � 7 MPa, α � 0.288 degree, K � 1, and l 2 � 0.2 mm. Te data can be run in MATLAB R2016b.

Conflicts of Interest
Te authors declare that they have no conficts of interest.