J. Cent. South Univ. (2014) 21: 140−150 DOI: 10.1007/s11771­014­1925­4

Flow field simulation and establishment for mathematical models of flow area of spool valve with sloping U­shape notch machined by

different methods

WANG Zhao­qiang(王兆强) 1 , GU Lin­yi(顾临怡) 1 , JI Hong(冀宏) 2 , CHEN Jia­wang(陈家旺) 1 ，LI Lin(李林) 1

1. State Key Laboratory of Fluid Power Transmission and Control (Zhejiang University), Hangzhou 310027, China;

2. School of Fluid Power and Control Engineering, Lanzhou University of Technology, Lanzhou 730050, China

© Central South University Press and Springer­Verlag Berlin Heidelberg 2014

Abstract: Precise function expression of the flow area for the sloping U­shape notch orifice versus the spool stroke was derived. The computational fluid dynamics was used to analyze the flow features of the sloping U­shape notch on the spool, such as mass flow rates, flow coefficients, efflux angles and steady state flow forces under different operating conditions. At last, the reliability of the mathematical model of the flow area for the sloping U­shape notch orifice on the spool was demonstrated by the comparison between the orifice area curve derived and the corresponding experimental data provided by the test. It is presented that the bottom arc of sloping U­shape notch (ABU) should not be omitted when it is required to accurately calculate the orifice area of ABU. Although the theoretical flow area of plain bottom sloping U­shape notch (PBU) is larger than that of ABU at the same opening, the simulated mass flow and experimental flow area of ABU are both larger than these of PBU at the same opening, while the simulated flow force of PBU is larger than that of ABU at the same opening. Therefore, it should be prior to adapt the ABU when designing the spool with proportional character.

Key words: spool valve; flow field simulation; flow area; steady state flow force; mathematical model; sloping U­shape notch

1 Introduction

There are different kinds of notches on the spool, such as U­shape, V­shape, K­shape and various U­V combinations, and some of them are very complicated in geometry. LU et al [1] presented that the major role of the notches was to enhance the controllability and durability of the valve, and in particular, to reduce the ripple pressure and the settling time in its dynamic performance. SHOHEI et al [2] thought that the hydraulic valve used in construction machinery features at notches on its spool was suitable for obtaining desired operation performance. Because of the special structure of the spool, the valve has different characteristics from the traditional spool valve.

In the last ten years, many researchers were active in the field of studying the influence of shape and number of spool notches on the discharge characteristics (discharge coefficient, velocity coefficient and flow angle) of a hydraulic distributor metering edge. JI et al [3] presented the computational results on the flow characteristics of U­shape and V­shape notches.

Three­dimensional models were developed with commercially available fluid flow models and simulated by computational fluid dynamics (CFD) softwares, and the flow features of flow rates, discharge coefficients, efflux angles and flow forces under different operating conditions were studied. YANG [4] presented some results from an attempt to characterize hydraulic oil flow inside spool valves with different spool metering notches using CFD, and explored the effects of geometric parameters of notches on important hydraulic oil flow characteristics, such as flow force and discharge coefficients. BORGHI [5] presented theoretical and experimental results of studies on the influence of shape and number of spool notches on the discharge characteristics. CAO et al [6] performed a valve testing to investigate the fluid field of spool valves with distributed circular ports. Valve ports with and without U­shape notches were tested to characterize the notch effect on both the fluid force and flow rate. At last, equivalent jet angle and discharge coefficient were calculated from the measurements based on the lumped parameter models. AMIRANTE et al [7] presented the evaluation of the driving forces acting on a 4/3 hydraulic

Foundation item: Project(51004085) supported by the National Natural Science Foundation of China Received date: 2012−08−20; Accepted date: 2012−11−20 Corresponding author: WANG Zhao­qiang, PhD Candidate; Tel: +86−751−87953028; E­mail: wangzhaoqiang_2008@zju.edu.cn

J. Cent. South Univ. (2014) 21: 140−150 141

open center directional control valve spool by means of a complete numerical analysis. In the valve body, a hollow sliding spool with four patterns of radial holes was visible. AMIRANTE et al [8] made a comparative evaluation of the main characteristic parameters (flow force, flow rate and efflux angle) between the hemispherical notch with the cylindrical notch and without it. A numerical method and an experimental analysis were applied by ZOU et al [9] to obtain a better understanding on the characteristics of cavitations inside the non­circular opening spool valve with U­grooves. The spool valves studied by DENG et al [10] were simplified from that used in a real servo system, in which a spool consists of several groups of notches, such as U­shape, K­shape and V­shape notches. It was shown that the combination of these different shape notches could lead to a specific flow rate pressure relation. JIA and YIN [11] presented the computational results of studies on the flow characteristics of V­U and U­U notches. Similarly, three­dimensional models were developed with commercially available fluid flow model in order to simulate the flow features. CHO et al [12] developed a kind of spool with various combinations of U­shape notches, which could reduce hydraulic energy losses at the main control valve on excavator by the optimal design of the opening areas for spools under the consideration of the maneuverability of the vehicle and pump characteristics. However, there were still dead zones at the orifice opening of 0−2 mm.

PAN et al [13] analyzed the discharge characteristics of servo­valve spool valve under the conditions of laminar and turbulent flow and established a general mathematical model for the orifice flow. The discharge characteristics of spool valve orifices were simulated by the CFD method and a formula for discharge coefficient and Reynolds number was derived.

Many researchers above have introduced various U­shape notches on the spool valve, however, the sloping U­shape notch machined by an end­milling cutter on the spool has not been investigated. The sloping U­shape notch has little dead zone and good linear. Currently, there is no exact function expression of the flow area of the sloping U­shape notch orifice in literatures, including in the software of Amesim R10. It is necessary to establish the function expression of flow area in order to analysis the hydraulic system by the software of Amesim R10. Detailed knowledge of the internal flow field of the sloping U­shape notch is required for the design optimization of it, CFD is becoming a well established practice in valve analysis and design [14−17].

Therefore, it is meaningful to present the mathematical model of the flow area for the sloping U­shape notch orifice machined by an end­milling cutter on the spool. In particular, it derives the precise function

expression of the flow area for the sloping U­shape notch orifice versus the spool stroke. In addition, to simulate the flow features, three­dimensional models are developed with commercially available fluid flow models. Then CFD is used to analyze the flow features of the sloping U­shape notch of valves, including the flow features of flow rates, discharge coefficients, efflux angles, and steady state flow forces under different operating conditions.

2 Structure of sloping U­shape notch on spool

Figure 1 shows the mathematical model of the sloping U­shape notch, which is simplified from a non­circular opening spool valve with four sloping U­shape notches. It is machined by end­milling cutter on CNC machine tools, according to the two methods shown respectively in Fig. 2.

Fig. 1 Mathematical model of sloping U­shape notch (Unit: mm)

Fig. 2 Methods for machining sloping U­shape notch: (a) Horizontal machining; (b) Sloping machining

When it is machined by the method shown in Fig. 2(a), the cross section view is shown in Fig. 3, its planform looks like a U letter. Its end is an ellipse, judging from its section view, its top is arc, its middle is rectangle and its bottom is arc. In this work, it is named

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arc bottom sloping U­shape notch (ABU). When machined by the method shown in Fig. 2(b), the cross section view is shown in Fig. 4, its planform looks like a U letter. Its end is also an ellipse, judging from its section view, its top is arc and its bottom is plain. In this work, it is named as plain bottom sloping U­shape notch (PBU). That is to say, although with the same drawing, but with different machining methods, its real structure is not same. Next, a further investigation is needed to study them respectively.

Fig. 3 Cross section view of arc bottom sloping U­shape notch

Fig. 4 Cross section view of plain bottom sloping U­shape notch

In order to analyze the flow area for the sloping U­shape notch conveniently, the spool orifice is presumed zero overlap when the spool is at the neutral position. As shown in Fig. 5, when hydraulic oil flows through the notch, it will flow through the cross section

Fig. 5 Orifice area for sloping U­shape notch

area As and the projection area Ap. The smaller one of them is the right flow area. With the orifice opening becoming big, the smaller flow area is either the projection area Ap on the spool, or the cross section area As in the notch. It is easy to know that the projection area Ap may be the flow area at a little opening, and the cross section area As must be the flow area at a big opening. The research only focuses on the cross section area As in the notch which is assumed to be the flow area of the sloping U­shape notch.

3 Function expression of flow area for sloping U­shape notch

The schematic diagrams of the A−A′ view and B−B′ view of ABU are established in Figs. 6 and 7, respectively, and the schematic diagrams of the A−A′ view and B−B′ view of PBU are established in Figs. 8 and 9, respectively.

3.1 Mathematical model for ABU orifice opening f≤x≤ L1

It is easy to know that the end of the planform of ABU is an ellipse, and the length of its semi­major axis is equal to the value of f, the length of its semi­minor axis is equal to r.

β

β β

sin

sin 2

) ( cos 2 d h R

f + −

= (1)

Based on the original point of n, the ellipse equations are given as

1 ) ( 2

2

2

2

= −

+ f x f

a y (2)

2

2) ( 1 2

f x f a L

− − = (3)

β cos ) ( 0 1 oab ab F A A A ∆ − = (4)

β cos 1 2 Lh A = (5)

β sin π 8 1 2

3 d A = (6)

3 2 1 1 s A A A A + + = (7)

The related equations are substituted into Eqs. (4), (5) and (6), then

2 2 1

1 2 2 2 1 25 . 0 5 . 0

2 arcsin

h L

L d R d R d R A

+

− − = (8)

− + − −

+ = 2 2

1 2 2 1

1 2 25 . 0 ) 5 . 0 ( d R R d x

L h

h L

dL A (9)

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Fig. 6 Schematic diagrams of ABU orifice opening f≤x≤L1

Fig. 7 Schematic diagrams of ABU orifice opening 0≤x<f

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Fig. 8 Schematic diagrams of orifice opening 1 L x f ′ = = ′

Fig. 9 Schematic diagrams of orifice opening 0=x=f′

2 2 1

2 3 π

8 1

h L

h d A +

= (10)

where A1 is the top fan area for flow area, mm 2 ; A2 is the middle rectangle area for flow area, mm 2 ; A3 is the bottom fan area for flow area, mm 2 ; AFoab is the fan area in right view, mm 2 ; A∆oab is the triangle area in right view,

mm 2 ; f is the length of semi­major axis of ellipse, mm; d is the diameter of end­milling cutter, mm; h is the depth of start­machining notch, mm; h1 is the height of the rectangle in the right view, mm; h2 is the distance from the top of rectangle to the center of spool, mm; L is the width of notch, mm; L1 is the length of notch, mm; R is the radius of spool, mm; x is the orifice opening, mm; y

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is the variable of the ellipse, mm; θ is the angle of cross­section fan, rad; β is the angle of sloping U­shape notch, rad.

3.2 Mathematical model for ABU orifice opening 0≤ x＜f

β cos ) ( 4 oab Foab A A A ∆ − = (11) β sin ) ( 5 ncm Fncm A A A ∆ − = (12)

5 4 s2 A A A + = (13)

The related equations are substituted into Eqs. (11) and (12), then

−

− − = 2

2 2

4 ) ( 1 arcsin

f x f

R a R A

⋅

− − −

− + − 4

4

2 2

2

2 2 2 2 ) ( ) ( 2 x f

f a x f

f R a a R a

2 2 1

1

h L

L

+ (14)

⋅ −

− − = a

f x f

d a d A 2

2 2

5 ) ( 1 2 arcsin 25 . 0

⋅

− − −

− + − 4

4 2

2

2 2 2 2 ) ( ) ( 25 . 0 2 25 . 0

f x f x f

f d a a d

2 2 1 h L

h

+ (15)

As a whole, the function expression of the flow area for ABU with the orifice change is given as

s1 1

s2

, , 0

A f x L A

A x f ≤ ≤

= ≤ < (16)

where A is the flow area of orifice, mm 2 ; A4 is the top fan area for flow area, mm 2 ; A5 is the bottom fan area for flow area, mm 2 ; As1 is the flow area of orifice opening f<x≤L1, mm 2 ; As2 is the flow area of orifice opening 0≤x<f, mm 2 ; AFoab is the fan area in right view, mm 2 ; A∆oab is the triangle area in right view, mm 2 ; AFncm is the fan area in planform view, mm 2 ; A∆ncm is the triangle area in planform view, mm 2 ; f is the length of semi­major axis of ellipse, mm; d is the diameter of end­milling cutter, mm; h is the depth of start­machining notch, mm; h2 is the distance from the top of rectangle to the center of spool, mm; L is the width of notch, mm; L1 is the length of notch, mm; R is the radius of spool, mm; x is the orifice opening, mm; θ is the angle of cross­section fan, rad; β is the angle of sloping U­shape notch, rad.

3.3 Mathematical model for PBU orifice opening f′= x=L′1 It is easy to know that the end of the planform of

PBU is also an ellipse, and the length of its semi­major axis is equal to f ′, the length of its semi­minor axis is equal to r′.

( ) 2 2 0.25 ) ctg f R R d β ′ ′ ′ ′ = − − (17)

Based on the original point of n′, the ellipse equations are given as

2 2

2 2 ( )

1 y f x a f

′ ′ ′ − + =

′ ′ (18)

2

2 ( )

2 1 f x L a f

′ ′ − ′ ′ = − ′

(19)

β ′ − = ′ ′ ′ ′ ∆ ′ ′ ′ cos ) ( 1 b a o b a o F A A A (20)

β ′ ′ ′ = ′ cos 1 2 h L A (21)

2 1 1 A A A s ′ + ′ = ′ (22)

The related equations are substituted into Eqs. (20) and (21), then

2 21

1 2 2 2 1 25 . 0 5 . 0

2 arcsin

h L

L d R d R d R A

′ + ′

′

′ − ′ ′ −

′ ′

′ = ′

(23)

′ − ′ + ′ −

′ ′ ′

′ + ′

′ ′ = ′ 2 2

1 2 21

1 2 25 . 0 d R R

L x h

h L

L d A (24)

where 1 A′ is the top fan area for flow area, mm 2 ; 2 A′ is the bottom rectangle area for flow area, mm 2 ; 1 s A′ is the flow area of orifice opening ,1 L x f ′ ≤ ≤ ′ mm 2 ; b a o F A ′ ′ ′ is the fan area in right view, mm 2 ; b a o A ′ ′ ′ ∆ is triangle area in right view, mm 2 ; f′ is the length of semi­major axis of ellipse, mm; d′ is the diameter of end­milling cutter, mm; h′ is the depth of start­machining notch, mm; 1 h′ is the height of the rectangle in the right view, mm; 2 h′ is the distance from the top of rectangle to the center of spool, mm; L′ is the width of notch, mm; 1 L′ is the length of notch, mm; R′ is the radius of spool, mm; x′ is the orifice opening, mm; y′ is the variable of the ellipse, mm; θ′ is the angle of cross­section fan, rad; β′ is the angle of sloping U­shape notch, rad.

3.4 Mathematical model for PBU orifice opening 0= x<f′

β ′ − = ′ ′ ′ ′ ∆ ′ ′ ′ cos ) ( 2 s b a o b a o F A A A (25)

As a whole, the function expression of the flow area for PBU with the orifice change is given as

′ < ′ ≤ ′

′ ≤ ′ ≤ ′ ′ = ′

f x A L x f A

A 0 , ,

2 s

1 s1 (26)

where 3 A′ is the top fan area for flow area, mm 2 ; 1 s A′ is the flow area of orifice opening ,1 L x f ′ ≤ ≤ ′ mm 2 ; 2 s A′ is the flow area of orifice opening , 0 f x ′ ≤ ≤ mm 2 ; b a o F A ′ ′ ′

J. Cent. South Univ. (2014) 21: 140−150 146

is the fan area in right view, mm 2 ; b a o A ′ ′ ′ ∆ is the triangle area in right view, mm 2 ; m c n F A ′ ′ ′ is the fan area in planform view, mm 2 ; A∆n′c′m′ is the triangle area in planform view, mm 2 ; f ′ is the length of semi­major axis of ellipse, mm; d ′ is the diameter of end­milling cutter, mm; h′ is the depth of start­machining notch, mm; 2 h′ is the distance from the top of rectangle to the center of spool, mm; L′ is the width of notch, mm; 1 L′ is the length of notch, mm; R′ is radius of spool, mm; x′ is the orifice opening, mm; θ′ is the angle of cross­section fan, rad; β′ is the angle of sloping U­shape notch, rad.

3.5. Flow area curve versus spool displacement In Fig.10, the orifice area curves for ABU and PBU

versus the spool displacement are shown, which are calculated by Matlab code.

Fig. 10 Orifice area curves for ABU and PBU notch

The flow area for ABU has good linearity, and the dead zone is little, which is very suitable for pilot proportional control. While the flow area for PBU does not has good linearity in the first part of the spool displacement (0−0.5 mm), and after that, the flow area has good linearity, which is also very suitable for pilot proportional control. The former is more suitable for micro controlling than the latter. At the same opening,

the flow area of PBU is larger than the flow area of ABU.

4 CFD simulation

Fluent code is used for calculations in this work. The high­pressure flow inside the valve is assumed to be not completely turbulence, and the realizable k­epsilon model is applied to all the simulations because of its numerical stability under a condition of large pressure gradient. The fluid density is 872 kg/m 3 , and the dynamic viscosity is 0.040112 kg/(m∙s). These correspond to a typical hydraulic fluid at 40 °C. The viscosity and density of the oil are all considered constant. The oil is Newtonian and incompressible. A segregate implicit steady state solver is used. The second order equation is selected in the solution control. Convergence criteria of 10 −5 for velocities and continuity are applied.

In Fig. 11, the boundary condition surfaces are indicated, and no slip velocity is taken as boundary condition.

Fig. 11 Computational grid in modeled geometry

1) Inlet: On this surface, pressure inlet boundary condition is respectively setup to the values of 4, 16 and 24 MPa, which are the real operational conditions.

2) Outlet: On this surface, the pressure outlet boundary condition is setup to 0.1 MPa constantly.

3) Symmetry: The symmetry faces are shown in Fig. 11.

On the other side, the grid number of about 3.59×10 5 is used. A particular technique provided by ICEM­CFD, the “size function”, allows the refinement of the grid near the metering sections with the maximal velocity and pressure gradients.

4.1 Mass flow In Fig. 12, the mass flow curves for ABU and PBU

are shown. Judging from Fig. 12, the mass flow for ABU is larger than that for PBU at the same opening and pressure difference, because the bottom for PBU is plain, and that for ABU is arc, the turbulence density for PBU

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Fig. 12Mass flow curves for ABU and PBU

is larger than that for ABU. And with the orifice becoming big, the mass flow for ABU increases linearly. With the pressure difference increasing, the mass flow will increase at the same opening. Moreover, with the orifice becoming big, the mass flow for PBU will not well increase linearly, which is not suitable for the proportional control system.

4.2 Steady state flow force In Fig. 13, the steady state flow force curves for

ABU and PBU are shown. Judging from Fig. 13, with the orifice becoming big, the steady state flow forces of both ABU and PBU increase linearly. With the pressure difference increasing, the steady state flow force increases at the same opening. However, the steady state flow force for PBU is larger than that for ABU at the same opening and pressure difference. That is to say, the ABU has good operation performance.

Fig. 13 Steady state flow force curves for ABU and PBU

4.3 Efflux angle The efflux angles through the metering section in

the three different pressures increase with the axial spool movement, as shown in Fig. 14. Moreover at the same

Fig. 14 Efflux angle through metering section under different pressures: (a) 4 MPa; (b) 16 MPa; (c) 24 MPa

opening, the efflux angles through the metering section in the three different pressures nearly keep constant. While in the case of the same opening and pressure difference, the efflux angle for ABU is larger than that for PBU. In the part of the opening (0−1 mm), the efflux angle for ABU is nearly zero, which is because the mass flow is nearly zero in the range of opening (0−1 mm). While in the part of the opening (0−1.5 mm), the efflux angle for PBU is nearly zero, which is because the mass flow is nearly zero in the range of opening (0−1.5 mm). It is evident that the dead zone for ABU is shorter than that for PBU.

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4.4 Flow coefficient The flow coefficients through the metering section

in the three different pressures increase with valve opening, as shown in Fig. 15. Moreover, at the same opening, the flow coefficients through the metering section in the three different pressures nearly keep constant except the case of 16 MPa. While the flow coefficient for ABU is larger than that for PBU in the case of the same opening and pressure difference. In the part of the opening (0−0.5 mm), the flow coefficients for

Fig. 15 Flow coefficient through metering section under different pressures: (a) 4 MPa; (b) 16 MPa; (c) 24 MPa

ABU and PBU are both nearly zero, because the mass flow is nearly zero in the range of opening (0−0.5 mm).

4.5 Velocity distribution The flow velocities through the metering section in

the three different pressures increase with spool displacement until it is up to the maximum value, as seen in Fig. 16. Moreover, at the same opening, the flow velocities through the metering section increase with the increasing pressure difference. While the flow velocity for ABU is larger than that for PBU in the case of the same opening and pressure difference. In the part of the opening (0−0.5 mm), the flow velocities for ABU and PBU are both nearly zero under 4 MPa.

Fig. 16 Velocity for ABU and PBU at different pressures

5 Test and analysis

5.1 Principle of test Figures 17 and 18 show the test scheme of the

sloping U­shape notch orifice. When screw 1 is adjusted, spool 3 moves gradually at the push of screw 1, then the orifice of spool 3 changes relatively. While dial indicator 7 can display the moving distance of spool 3, the presure can be achieved from pressure transducer located at the outlet and inlet, and the flow rate can be measured by the flowmeter located at the outlet.

As for the orifice whose turbulent losses are dominant, the hydraulic resistance is described by its geometry [18]. The flow area A at the orifices is calculated as

ρ p C Q A

∆ = 2 / d (27)

where Cd=0.62, ρ=872 kg/m 3 , ∆p=4 MPa. The orifice area of the spool can be obtained.

The measurement errors for valve performance are estimated. The error of the structural parameters is ±1%, flow rate is ±0.5%, pressure is ±1%, temperature is ±0.5% and opening is ±0.2% [9].

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Fig. 17 Test scheme of orifice

Fig. 18 Photo of test appliance of orifice

The related parameters of the orifice tested are shown in Fig. 1, and the diameter of the spool with four sloping U­shape notches is 14.5 mm. The flow areas for ABU and PBU are tested, respectively.

5.2 Analysis The theory data and the experimental data of the

orifice area at different orifice openings are shown in Figs. 10 and 19, respectively. The point curve shows the experimental data of ABU with the orifice opening change, and the triangle curve shows the experimental data of PBU with the orifice opening change in Fig. 19. The dashed curve shows the theoretical data of ABU with the orifice opening change, and the solid curve shows the theoretical data of PBU with the orifice opening change in Fig. 10.

The theory data in Fig. 10 is in reasonable agreement with the experimental data in Fig. 19, and the maximum error is up to about 8%, which can be accepted if the leakage of the spool, measurement error of flow and the section factor coefficient of the orifice are taken into consideration.

6 Conclusions

1) The two mathematical models on the flow area of the sloping U­shape notch orifice machined by an end­ milling cutter on the spool valve are both presented and analysed. The validity of the two mathematical models is

Fig. 19 Experimental data for ABU and PBU flow area

proven by comparing it with the experimental data. 2) It is verified that the bottom arc for ABU should

not be omitted when it is required to accurately calculate the orifice area of the ABU. When ABU opening is 0≤ x＜f, the nearly linear relationship exists between the flow area for ABU and the orifice opening. Moreover, when PBU opening is 0≤x<f′, the curve relationship exists between the flow area for PBU and the orifice opening. The output flow for ABU and PBU can be both linearly proportional to the spool displacement under constant pressure drop at a larger opening. Therefore, the former is more suitable for micro controlling.

3) Although the theoretical flow area of PBU is larger than that of ABU at the same opening, the simulated mass flow and experimental flow area of ABU are both larger than these of PBU at the same opening. While the simulated flow force of PBU is larger than that of ABU at the same opening. Therefore, it should be prior to adapt the ABU when designing the spool with proportional character.

4) In the further research, the flow force and temperature distribution in the sloping U­shape notch should be also measured through a test. Moreover, the performance of pilot proportional controlling and micro controlling should be also tested.

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(Edited by FANG Jing­hua)