- Email: [email protected]
Effects of groove shape of notch on the flow characteristics of spool valve
Energy Conversion and Management 86 (2014) 1091–1101
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Effects of groove shape of notch on the flow characteristics of spool valve Yi Ye, Chen-Bo Yin ⇑, Xing-Dong Li, Wei-jin Zhou, Feng-feng Yuan Institute of Automobile and Construction Machinery, Nanjing Tech University, Nanjing 211816, China
a r t i c l e
i n f o
Article history: Received 23 February 2014 Accepted 26 June 2014 Available online 15 July 2014 Keywords: Flow characteristics Spool valve Notch Groove shape Computational fluid dynamics (CFD)
a b s t r a c t The grooves of notches of hydraulic spool valves are usually designed into various shapes for their desired flow characteristics. The aim of this paper is to clarify the effects of the groove shape on the flow characteristics through computational fluid dynamics (CFD) and experimental investigations. The RNG k–e turbulence model is used to simulate the pressure distributions of the flow fields inside three notches with their corresponding typical structural grooves in order to analyze the changes of restricted locations along with the openings and, furthermore, to calculate the flow areas of the notches. The accuracy of the employed model is demonstrated by comparing the computational results with the experimental data. Additionally, the flow rate vs. pressure drop data obtained from the experiment is fitted by least square method. On this basis, the discharge coefficient as a function of groove geometry, flow condition, fitting coefficients and its stable value is deduced, proving to be quite consistent with the experimental result. Thanks to the jet flow angles estimated by CFD simulation, the steady flow forces are calculated, which show good agreement with the experimental results except for some small differences. Finally, the throttling stiffness of the three notches is investigated, with that of divergent U-shape groove falls between spheroid-shape groove and triangle-shape groove. Similar results are found for steady flow force. The results indicate that the groove shape has significant effects on the flow characteristics (flow area, discharge characteristic, jet flow angle, steady flow force and throttling stiffness) of spool valve. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Work performance of a hydraulic system is largely determined by the hydraulic valve. And fluid flow control of the hydraulic valve essentially depends on the throttling form of notches. Notches with throttling grooves are widely used in high performance hydraulic valves where excellent precision and stability are required due to the following advantages, such as diverse structure, wide range of flow rate, the gradient of flow area being easy to control, good stability in the case of small flow, abundant flow control characteristics. Some scholars have launched a number of pioneering researches to study the relationship between the valve geometry and its flow characteristics. Lisowski et al. [1] proposed a logic type directional control valve with reduction of pressure losses over 35%, while the difference taken from CFD analysis and experimental research does not exceed 5%. They [2] also proposed the calculation of flow forces (pressure force and viscous force) acting on the spool of solenoid operated directional control valve using 3D CFD modeling along with k–e turbulence model. Pan et al. [3] analyzed the discharge
⇑ Corresponding author. Tel./fax: +86 025 58139599. E-mail address: [email protected] (C.-B. Yin). http://dx.doi.org/10.1016/j.enconman.2014.06.081 0196-8904/Ó 2014 Elsevier Ltd. All rights reserved.
characteristics of servo-valve spool valve under both laminar and turbulent flow by the CFD method and derived a formula for discharge coefficient and Reynolds number. Posa et al. [4] carried out an analysis of the discharge coefficient and the flow force of a directional valve utilizing 2D CFD method and, in addition, the fluid-body interaction had been represented by an immersedboundary technique. Amirante et al. [5] performed a complete 3D numerical simulation to deal with the driving forces acting on an open center directional control valve. Yuan and Li [6] highlighted the important effects of two often ignored components, viscosity effect and non-metering momentum flux, on the steady flow force for the purpose of achieving outstanding dynamics performances. All the above researches have demonstrated that the CFD method has advantages to predict flow characteristics of valve and is efficient in capturing many partial details in flow field. Moreover, an experimental procedure had been proposed to determine the discharge coefficient of a spool valve as a function of opening and pressure drop in Ref. [7]. Ji et al. [8] developed a pressure distribution and noise measurement test equipment of hydraulic valve with a mobile valve sleeve, and discussed the interior relations among the throttling groove structure, pressure distribution and noise characteristics. Furthermore, relevant works with reference to flow area, flow coefficient and steady flow force of spool valve
1092
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
with notches were experimentally investigated in other studies of Ji et al. [9–11]. Some other notable works aimed at flow characteristics include studies on flow test of ball valve by He et al. [12], dynamic analysis of a two-stage solenoid valve flow by Ye et al. [13], flow structure inside a spool type pressure regulating valve by Chattopadhyay et al. [14] and visualization experiment of flow patterns and cavitation phenomena by Chern et al. [15]. In general, these researches were focused on global performances of valves. Unfortunately, the influences of valve geometry on flow characteristics were not highlighted. Some investigations proposed functional structures achieved through modifying or changing the geometry of valve to improve flow force balance, anti-cavitation ability, flow capacity, etc. Amirante et al. [16] numerically studied the effect of optimized groove and compensation profile on reducing the flow force of a proportional valve, and also provided the corresponding experimental verification. Borghi et al. [17] presented three different valve port connections which were used to provide compensation to stationary axial flow forces of a reference spool valve, and the compensation effect was verified by experiment. Many other innovative designs were also presented, for instance, arc notches in spool land and sleeve [18], additional parallel and compensatory channels in a solenoid operated directional control valve [2], tapered structure on spool rod [19] and rotated channel in non-metering port [20]. And positive effects of the novel structures in the above articles were also shown. However, limited work [21–23] has been carried out to study the influence of groove shape on flow characteristics although groove shape, to a great extent, may determine the observable behavior of valve. Moreover, the effects of groove shape on flow characteristics still lack systematic research. By investigating the effects of groove shape on flow characteristics of spool valve from multiple perspectives, this study fills up the gap of previous studies by proposing three types of throttling grooves with different structural features, namely spheroid-shape groove, triangle-shape groove and divergent U-shape groove. Following a brief description of groove structures, the pressure distributions of the flow fields inside the notches are numerically simulated so as to analyze the changes of restricted locations along with the increasing spool opening and to calculate the equivalent notch area. Then, a least square method is utilized to fit the test data of flow rate vs. pressure drop. The discharge coefficient as a function of groove geometry, flow condition, fitting coefficients and its stable value is deduced, and stable values of the discharge coefficient are obtained and discussed. After that, the steady flow force is calculated as the jet flow angle is estimated by CFD numerical simulation. Finally, the throttling stiffness of the notch is also investigated. The results provide effective guidance for the design of high-performance hydraulic valve. 2. CFD analysis 2.1. Groove structures The three types of throttling grooves with typical structural features are shown in Fig. 1. Fig. 1a is a spheroid-shape groove and Fig. 1b is a triangle-shape groove. Fig. 1c is a divergent U-shape groove with a relatively complex structure, which could be considered as being overlaid in parallel by a U-shape part and a divergent one. In Fig. 1, X is the spool opening; A1 and A2 are the axial and radial cross-sections respectively, and both cross the throttling edge. A3 is the cross-section which crosses both the throttling edge and the lowest point of the groove. Amin is the smallest cross-section across the throttling edge and which is usually perpendicular to the bottom. To demonstrate the effects of the structural features on the flow characteristics, the lengths of the throttling grooves in
the axial direction are set to be equal (3 mm). To reduce the impact of radial imbalance force on the experimental results, two throttling grooves are symmetrically processed in the circumferential direction on the spool. 2.2. Modeling and simulation Control of fluid flow by notches is accomplished through regulating the opening so as to change the flow area. Therefore, determining the relationship between the opening and flow area is the foundation of carrying out the flow characteristics investigation. Generally, several cross-sections in the notch contribute to the throttling effect, resulting in complex flow structure and pressure distribution of the fluid field inside the notch. For single-stage throttling groove in the previous studies [9,10,23,24], throttling effect was usually attributed to axial, radial or minimum cross-section across throttling edge (defined as throttling cross-section thereof). But in fact, as the opening changes, the throttling crosssection varies dynamically. According to the Bernoulli law, flow velocity will rise sharply along with a rapid pressure drop when fluid flow through the throttling cross-section. In other words, a rapid pressure drop only happens in the throttling cross-section of the notch. Thus, in order to obtain the accurate location of the throttling cross-section, CFD numerical simulation method is applied to analyze the pressure distribution inside the notch. The grid model is established using half part of the flow field inside the valve for it is an axis symmetrical structure. As shown in Fig. 2 (take spheroid-shape groove in X = 0.6 mm as an example), it is divided by the application of tetrahedral mesh which contains about 450,000 cells. Grid models for triangle-shape groove and divergent U-shape groove cases contain about 435,000 and 455,000 cells, respectively. What’s more, it can be seen from Fig. 2 that the grid gradually becomes intensive in the process of getting close to the wall and the inlet and outlet boundaries of the notch. The RNG k–e turbulence model is selected, and boundary conditions of inlet and outlet pressures are imposed. In addition, solution accuracy is set to be 105, and hydraulic oil with viscosity of 46 cSt and density of 896 kg/m3 is chosen as the medium. Gridindependent test of the model is carried out with various grid sizes to make sure the obtained numerical solutions are independent on grid density. Five positions (X = 0.6, 1.4, 2.0, 2.8, 3.1 mm) during opening process are selected to be analyzed. 2.3. Pressure distribution The pressure distributions in the symmetrical surface of the notches at the corresponding positions are shown in Fig. 3. The pressure drop through the spheroid-shape groove is concentrated on cross-sections A2 and A3 within the entire range of the spool stroke. In other words, the throttling effect of the spheroid-shape groove is provided by cross-sections A2 and A3 together, showing two-level throttling characteristics. In particular, with the increase of the opening, the proportion of the pressure drop changes gradually from cross-section A2 to A3. The triangle-shape groove is a typical divergent structure, whose pressure distribution characteristics are relatively simple. In the triangle-shape groove, cross-sections A1 and A2 do not have throttling function. The pressure drop is consistently centralized in cross-section Amin. The divergent U-shape groove has a more complex behavior of the pressure distribution than the other two types. Migration of throttling cross-sections exists in this type as shown in Fig. 3. At small openings, pressure drop is consistently centralized in crosssections A1 and A2. In addition, even though cross-section A2 bears more pressure drop, two-level throttling characteristics dominate in the pressure distribution. With the increasing opening, the
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
1093
Fig. 1. Geometric structures of: (a) the spheroid-shape groove; (b) the triangle-shape groove and (c) the divergent U-shape groove.
cross-section A2 and inclined cross-section A3 caused by simplified treatment are both verified to be less than 4%, demonstrating its feasibility. MATLAB is used to develop the flow area calculation procedure of the notch. When the flow area curves are drawn, A/ Amax and X/Xmax are respectively set as ordinate and abscissa as shown in Fig. 4. In addition, A1, A2, A3 are the area sizes of A1, A2, A3, respectively. As shown in Fig. 4a, Aeq is the minimum and should be regarded as the notch flow area value of the spheroid-shape groove during the entire opening process. In the first half of the spool stroke, Aeq is almost equal to A2 for A2 is much smaller than A3. However, A2 and A3 contribute more equally to Aeq during the second half of the spool stroke as the ratio of A3 to A2 gradually decreases to 1. For the triangle-shape groove (Fig. 4b), the ratio of A1 to A2 remains constant. In this case, the notch flow area is set to the value of Aeq, which is also equal to Amin. With regard to the divergent Ushape groove, there is an intersection point in the spool opening at X = 2.0 mm which is approximately 0.667 in term of X/Xmax from Fig. 4c. When X is smaller than 2.0 mm, the notch flow area equals Aeq. When X is larger than 2.0 mm, it changes to the minimum throttling cross-sectional area Amin. Fig. 2. Grid model for the spheroid-shape groove case.
throttling effect of cross-section A2 declines gradually. Meanwhile, the pressure loss at cross-section A1 migrates to cross-section Amin. The migration process is still characterized by the two-level throttling. Later in the remaining opening process, as the area of crosssection A2 continually increases, the throttling effect of cross-section A2 decreases until it disappears. At the same time, single-level throttling feature appears and Amin becomes the centralized crosssection for the total pressure drop.
2.4. Flow area analysis Currently, in the study of flow area of the notch with throttling groove, an effective approach is applied by replacing multiple throttling cross-sections with an equivalent throttling cross-section. In present paper, the minimum value between Amin and Aeq is taken as the notch flow area where Amin and Aeq are the area sizes of Amin and the equivalent throttling cross-section, respectively. Furthermore, Aeq is set to the value of the equivalent area of A2 and A3 in series in the spheroid-shape groove and A1 and A2 in series in the other two types according to the analysis of pressure distribution in section 2.3. In fact, the top surface of the throttling groove is a circular surface. The boundary lines of the enclosed surface are space curves. Therefore, obtaining the specific analytical solution of the flow area by applying numerical integration method or solving nonlinear function becomes really complicated [24]. As such, the circular surface is simplified to a plane when the flow area is calculated. While the spheroid-shape groove has the maximum circumferential span among the three throttling grooves, the area errors of its radial
3. Experimental setup A sketch of the notch flow characteristics experimental setup is presented in Fig. 5. The inlet and outlet pressures of the notch are respectively adjusted by a variable pump (range 0–150 l/min, class 0.2 absolute) and a counterbalance valve. The maximum pressure of the system is set to be 30 MPa by a safety valve. The mechanical oscillation of the testing apparatus is effectively suppressed by the hose and filter, and so is the interference caused by the pulsation of the fluid flow. The movement of the valve spool is adjusted by a ball screw which is driven by a stepping motor. Two pressure sensors (range 0–60 MPa, class 0.2 absolute) are arranged on the valve chamber. The rate of flow through the tested notch is measured by a flow meter (range 0–150 l/min, class 0.1 absolute). The axial force acting on the spool is measured by a force sensor (range 0–150 N, class 0.1 absolute). A compensation method [25] is used to test the steady flow force. Also, hydraulic oil with viscosity of 46 cSt and density of 896 kg/m3 is selected as the working fluid in the test. Considering that the test result is greatly influenced by the form error of the spool structure (such as the circular bead of the groove) when the spool opening is at the limits, the opening range of 0.4 mm to 2.8 mm is set as the scope of study. What’s more, three feature positions of X = 0.6, 1.4, 2.2 mm are selected to be emphatically analyzed within the range. 4. Results and discussion 4.1. Discharge characteristics MATLAB is used to deal with the obtained data from the test. The curves of the flow rate vs. pressure drop when the openings
1094
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
Fig. 3. Pressure distributions inside the notches at positions: (a) X = 0.6 mm; (b) X = 1.4 mm; (c) X = 2.0 mm; (d) X = 2.8 mm and (e) X = 3.1 mm.
of the three notches are at the feature positions are shown in Fig. 6. It is worth noting that the scale of the horizontal axis of the flow rate is different in different plots. From Fig. 6, it is obvious that the spheroid-shape groove has the strongest flow capacity,
followed by the divergent U-shape groove and then the triangleshape groove under the same pressure drop. In addition, the corresponding curves of CFD simulation results are also provided in Fig. 6. A comparison between the experimental and the CFD
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
1095
Fig. 5. Sketch of the experimental setup: (1) variable pump; (2) counterbalance valve; (3) safety valve; (4) hose; (5) filter; (6) stepping motor; (7) ball screw; (8) valve spool; (9) pressure sensor; (10) flow meter; (11) slid platform and (12) pedestal.
Dp ¼ a1 Q 2 þ a2 Q
ð1Þ
The values of coefficients a1, a2 are presented in Table 1. All the values of the fitting regression coefficients R are higher than 0.994, indicating the fitted curves of the flow rate vs. pressure drop are in good agreement with the experimental data (Fig. 6). The fluid flowing through the throttling groove can be described by the orifice flow formula:
Q ¼ Cd A
sffiffiffiffiffiffiffiffiffi 2Dp
ð2Þ
q
where Cd is the discharge coefficient, A is the notch flow area, q is the density of the fluid. As shown in Eq. (2), the discharge characteristic of the notch is determined by parameters Cd, A, q and Dp. Cd and A are inherent properties of the notch, which are only related to the throttling groove shape. The notch flow area A reflects the flow capacity of the notch. Greater A means stronger flow capacity. Cd reflects the resistance characteristic of the notch. The greater Cd is, the smaller the flow resistance of the notch is. When the pressure difference between the inlet and outlet of the flow field is certain, the rate of flow through the notch becomes larger as A and Cd increase. Eq. (2) can be rearranged as
Dp q ¼ Q 2 2C 2d A2
ð3Þ
And likewise, the ratio of Dp to Q2 can be expressed from Eq. (1) as follows
Dp a2 ¼ a1 þ Q Q2 Fig. 4. Plot of the flow area vs. opening for: (a) the spheroid-shape groove; (b) the triangle-shape groove and (c) the divergent U-shape groove.
simulation results clearly indicates that, in each case the dependence of the flow rate on the pressure drop is not exactly the same yet very close. The differences between the test results and their corresponding simulation values do not exceed 7%. The distribution of the flow rate vs. pressure drop data conforms to a quadratic curve with its vertex being exactly the original point. Therefore, curve fitting of the data obtained from the experiment is conducted by the least square method. The fitting relation of the pressure drop Dp and the flow rate Q can be expressed as:
ð4Þ
Since the fluid medium is selected and the groove shape as well as the opening is certain, the corresponding parameters q, A can be determined. And a1, a2 are known from Table 1. Combining Eqs. (3) and (4), the discharge coefficient Cd as a function of Q can be expressed as
Cd ¼
1 A
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
2ða1 þ a2 =Q Þ
ð5Þ
When increasing flow rate Q continually to infinity, the discharge coefficient will gradually approach a stable value (Cdst):
C dst ¼ lim C d ¼ Q!1
1 A
rffiffiffiffiffiffiffiffi
q
2a1
ð6Þ
1096
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101 Table 1 Polynomial coefficients for data interpolation. Notch structure
Opening (mm)
a1 (MPa/(l/ min)2)
a2 (MPa/(l/ min))
Spheroid-shape groove
0.6 1.4 2.2
2.573e2 2.647e3 9.170e4
7.299e03 7.397e04 1.700e03
Triangle-shape groove
0.6 1.4 2.2
2.083e + 1 7.605e1 1.347e1
8.606e01 9.459e02 2.654e02
Divergent U-shape groove
0.6 1.4 2.2
1.168e1 2.103e2 1.058e2
3.962e03 5.121e03 2.785e03
Table 2 Stable value of the discharge coefficient. Notch structure
Opening (mm)
Cdst
Spheroid-shape groove
0.6 1.4 2.2
0.747 0.682 0.620
Triangle-shape groove
0.6 1.4 2.2
0.720 0.692 0.666
Divergent U-shape groove
0.6 1.4 2.2
0.651 0.646 0.652
The relation between the discharge coefficient and the flow rate is described by Eq. (7). However, it cannot tell the change of the discharge coefficient with the flow condition. Therefore, Reynolds number (Re) of the fluid flowing through the notch is introduced to characterize the flow condition and defined as:
Re ¼
mqDh l
ð8Þ
where m is the flow velocity, which can be expressed as m = Q/2Amin, and l is the dynamic viscosity of the hydraulic oil. Dh is the hydraulic diameter, which is defined as four times the ratio of wet crosssectional area to wet perimeter length. In this study, the flow regime of the flow field inside the notch is determined by the maximum flow velocity which occurs in the cross-section with the minimum area. Therefore, to calculate the Reynolds number the values of the wet cross-sectional area and wet perimeter length are got in cross-section Amin. Thus, Dh is expressed as
Dh ¼
4Amin C
ð9Þ
where C is the perimeter length of Amin. Combining Eqs. (8) and (9), the relation between the flow rate and the Reynolds number can be obtained: Fig. 6. Curves of the flow rate vs. pressure drop of: (a) the spheroid-shape groove; (b) the triangle-shape groove and (c) the divergent U-shape groove.
Q¼
lCRe 2q
ð10Þ
Substitute Eq. (10) into Eq. (7), and after arranging we have
In this case, the fluid flow through the notch is fully turbulent. The stable values of the discharge coefficient at the feature positions of the three notches are calculated using Eq. (6) and the results are shown in Table 2. Put Cdst into Eq. (5), it becomes:
1 Cd ¼ A
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ C dst 1 þ a2 =a1 Q 2a1 ð1 þ a2 =a1 QÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
ð7Þ
C d ¼ C dst
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 2qa2 =a1 lCRe
ð11Þ
Therefore, the discharge coefficient is a function of groove geometry, flow condition, fitting coefficients and its stable value. Substitute the values of Cdst, a1, a2, l, q and C into Eq. (11), plots of the dependences of the discharge coefficients on the Reynolds number are obtained for different notches at the feature openings as shown in Fig. 7. Re0.5 is set as the abscissa and its scale is different in different plots. After data processing of the test results using
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
1097
orifice flow formula directly, we can see that the calculated results are in good agreement with the experimental ones. It is worth noting that in all samples the discharge coefficients first increase rapidly with regard to Reynolds number in the laminar flow region, and then gradually achieve the stable values around the transitional zones which range from 10 to 30 in terms of Re0.5. As the throttling groove is narrow, the fluid flowing through the notch is quite fast and seriously constrained by its wall, leading to a constant change of flow direction. In this case, laminar flow is easy to be damaged as viscous forces between liquid particles weaken. Consequently, inertia force of the liquid plays a dominant role, resulting in a transitional zone in advance. With the increasing opening, the discharge coefficient of the spheroid-shape groove reaches the stable value Cdst more laggingly, while the triangleshape groove and the divergent U-shape groove show no obvious change. The relation between the stable value of the discharge coefficient and the opening is significant for designing hydraulic valves with diverse flow control functions. Within the opening range of 0.4–2.8 mm, the discharge characteristic test is performed every 0.2 mm. The pressure difference between the inlet and the outlet of the notch is set large enough to obtain the stable value of the discharge coefficient in each test. Substitute the test data and the notch flow area into Eq. (2), we obtain the discharge coefficient distribution with the opening as shown in Fig. 8. From the trend of the curve, the impact of the opening on the discharge coefficient could be summarized as follows: (1) The discharge coefficient of the spheroid-shape groove has an approximately negative linear relationship with the spool opening. At small openings, A3 A2, radial cross-section A2 plays the main throttling role. And the flow resistance of the throttling groove is small, meaning a relatively large Cd. With the increase of the opening, A2 increases rapidly. The area ratio of A3/A2 gradually decreases to 1 as the twolevel throttling feature becomes obvious and Cd decreases continuously. (2) When the spool opening X < 0.8 mm, the discharge coefficient of the triangle-shape groove increases quickly from 0.639 to 0.71 as a result of clogging which often occurs in fluid flow through the notch under the condition of small flow area and large pressure difference. On the contrary, it keeps decreasing when X > 1 mm. This phenomenon is mainly attributed to the occurrence of cavitation in the partial low-pressure zones of the throttling cross-section, which has been observed inside a similar notch with V-shape groove by visualization experiment [21]. The occurrence of cavitation decreases the actual flow area of the notch. The influence becomes more significant as the opening increases under a constant pressure drop which leads to a continuous decline of the discharge coefficient. (3) As for the divergent U-shape groove, when X < 1 mm, the discharge coefficient drops quickly from 0.683 to 0.587. As X > 1 mm, the discharge coefficient increases gradually. When X = 1.8 mm, Cd increases to 0.648 and remains almost unchanged thereafter. According to the flow area characteristics, in the initial opening process, the pressure drop is mostly concentrated in the cross-section A2. And the flow resistance of the throttling groove is mainly caused by A2, showing single-level throttling characteristic. With the increase of the opening, A2 increases drastically. At approximately X = 1 mm, the total pressure drop distributes equally in the cross-sections A1 and A2 as A1/A2 approaches 1, resulting in the maximum throttling effect during the entire opening process. As a consequence, Cd decreases to its lowest
Fig. 7. Plots of Cd vs. Re0.5 for: (a) the spheroid-shape groove; (b) the triangle-shape groove and (c) the divergent U-shape groove.
value. Thereafter, A2 continues to increase drastically. As the difference of A1 and A2 becomes larger, the two-level throttling characteristic weakens. When the opening exceeds 2 mm, the throttling effect is solely caused by Amin, showing single-level throttling characteristic.
1098
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
Fig. 9. Schematic diagram of fluid flow through the notch.
Fig. 8. Discharge coefficient distribution with the opening.
Based on the above analysis, we can see that the three throttling grooves can be applied to the hydraulic system for different purposes according to their corresponding discharge characteristics. At small openings, the discharge coefficient of the spheroid-shape groove is great, which is good at quickly building up the working pressure for the applied hydraulic system. The discharge coefficient of the triangle-shape groove decreases gently in relation to the opening after its initial rise, which is beneficial to fine tuning of the flow rate and smooth running of the hydraulic system. As to the divergent U-shape groove, the discharge coefficient is big when the opening is small and followed by a rapid decline, which is advantage to avoiding the vibration occurred during startup process of the hydraulic system. 4.2. Steady flow force Flow force is one of the key factors that determine the pros or cons of the hydraulic control valve performance. It not only affects the operating force on the spool, but also causes the self-excited vibration of the components of the hydraulic system. Accurate calculation of the flow force is the prerequisite for control accuracy and force balance of the spool. As shown in Fig. 9, the control volume under flow force is the region surrounded by spool wall, notch horizontal surface and valve body underside. According to the momentum conservation law, the axial force balance equation for the control volume can be expressed in vector form (positive to the right) as follows:
~ F land þ ~ F body þ ~ F rod þ qQ ð~ v1 ~ v2Þ ¼ 0
As the area of the outlet is considerably larger than that of the inlet, the outlet velocity v2 is much smaller than the inlet velocity v1. Furthermore, the downstream flow is constrained by the spool wall with the outlet flow angle h2 being close to 90°. As a consequence, the outlet efflux force, namely the second term on the right side of Eq. (14) can be ignored. According to the literature [18,20], the viscous force can be neglected because it is too small comparing with the inlet efflux force (the first term on the right side of Eq. (14)), i.e., Fbody = 0. It’s worth noting that the inlet velocity v1 is not the jet flow velocity at the notch, but the velocity on the inlet surface (i.e. A2) of the control volume. By substituting Eq. (2) into Eq. (14), steady flow force can be further expressed as:
Ff ¼
2C 2d A2 cot h1 Dp A2
ð15Þ
Table 3 Jet flow angles at variable openings. Opening (mm)
Jet flow angle (°) Spheroid-shape groove
Triangle-shape groove
Divergent U-shape groove
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
78.5 74.8 72.0 70.3 69.1 67.9 67.2 66.6 66.2 65.7 65.3 64.9 64.6
44.6 45.2 44.2 43.9 43.8 42.5 43.4 44.4 44.5 44.2 43.9 44.5 43.6
67.0 66.4 66.1 65.8 65.3 64.9 64.1 63.4 63.0 62.7 62.5 62.2 61.1
ð12Þ
~ F land is pressure force vector subjected by the spool lands; ~ F body is F rod is viscous viscous force vector subjected by the valve body; ~ force vector subjected by the spool rod; ~ v 1 and ~ v 2 are inlet velocity F body vector and outlet velocity vector, respectively. The direction of ~ F rod are toward the left as the fluid flow to the right, which can and ~ be judged easily from Fig. 9. F f that the spool experiences from fluid The steady flow force ~ flow is equal to the sum of pressure forces ~ F land0 acting on the spool lands and viscous force ~ F rod0 acting on the spool rod, which is given in vector form as:
~ Ff ¼ ~ F land0 þ ~ F rod0
ð13Þ
~ F land and ~ F land0 are equal but in the opposite direction, the same as ~ F rod0 . The above two equations lead to the following scalar F rod and ~ form:
F f ¼ qQ v 1 cos h1 qQ v 2 cosh2 F body
ð14Þ
Fig. 10. Steady flow force at variable openings.
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
When the opening is certain, Cd, A, A2, h1 are fixed values. Assuming coefficient Ks = 2Cd2A2coth1/A2 is constant, the steady flow force Ff is proportional to pressure drop Dp. Flow structure
1099
in the control volume cannot be observed in the experiment, so CFD simulation is used to determine the jet flow angle h1. Table 3 summarizes the jet flow angles of the three throttling grooves
Fig. 11. Contours of pressure and velocity distributions of the divergent U-shape groove at opening: (a) X = 0.6 mm; (b) X = 1.4 mm and (c) X = 2.2 mm.
1100
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
at different openings. The jet flow angle variation range of the spheroid-shape groove reaches 14°, with great changes at the initial openings. In the intermediate positions, it is very close to the theoretical value of 69° derived by Von Mises. The jet flow angle of the triangle-shape groove is significantly affected by the flowdirection guide of the bottom slope and independent of the opening. Throughout the opening, the jet flow angle is small, and maintains at about 44°. The jet flow angle of the divergent U-shape groove decreases roughly linearly from 67° to 61.1°. It is noteworthy that for the steady flow force experiment, the pressure difference between the inlet and the outlet is set to 20 MPa for each test, with the error range of ±0.5 MPa, and the same pressure difference is applied for the corresponding calculation. Fig. 10 shows the comparison between the calculated and experimental values of the steady flow force at different openings. It can be seen that theoretical calculations can fully reflect variation trends of the experimental steady flow forces. The steady flow forces of the triangle-shape groove and the divergent U-shape groove increase roughly linearly with the increasing opening. At very small openings, the steady flow force of the triangle-shape groove appears to be negative, tending to open the valve. This appearance may be because of the occurrence of complex vortex and reflux in the exit section of the control volume, which makes the axial exit momentum non-ignorable or even larger compared to the axial entry momentum. At the same time the axial entry momentum is very tiny as the discharge coefficient and the notch flow area are both at a low level. Similar phenomenon was also observed in the experimental investigation of spool valve with Ushape grooves at small openings [22]. At the same opening, the test values are slightly smaller than the calculated values for all the three cases. The differences gradually increase with the increasing openings. A hypothesis has been made for the calculation of the steady flow force that the pressure difference between the inlet and the outlet of the flow field is considered wholly concentrated on the throttling section. This hypothesis actually presents a larger pressure drop through the throttling section in the calculation than the test, leading to the differences between the calculated values and the test values. Fig. 11 provides the velocity and pressure distributions in the symmetrical plane at opening X = 0.6, 1.4, 2.2 mm of the divergent U-shape groove under the same pressure drop of 5 MPa. At X = 0.6 mm, the flow velocity attains its maximum value inside the groove region, which means the pressure drop is mostly restricted to the throttling section as shown in Fig. 11a. With the increase of the opening, the region of the maximum velocity migrates downstream of the control volume. Correspondingly, the pressure losses dissipated at non-throttling sections become more and more significant with reference to the main pressure drop through the throttling groove as can be seen in Fig. 11b and c. Similar results have been obtained for the other two types (data not shown). In addition, with the increase of the opening, the outlet area reduces and the outlet flow angle gradually deviates from 90°due to diversion by the spool shoulder, both causing a slight increase of the outlet efflux force. For these reasons, the differences between the calculated values and the corresponding test values of the steady flow forces gradually enlarge with the increasing opening. The same reasons apply to the spheroid-shape groove as well as the triangle-shape groove. When the notch is close to fully opened status, the flow resistance of the throttling section virtually disappears. And the test values of the steady flow force increase suddenly while the flow rate increases rapidly (Fig. 10). 4.3. Throttling stiffness When the discharge of the notch extremely depends on the external load, minor change of pressure difference will still cause
Fig. 12. Plot of throttling stiffness with variable pressure drops and openings for: (a) the spheroid-shape groove; (b) the triangle-shape groove and (c) the divergent U-shape groove.
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
severe discharge oscillation, leading to instability of the flow control. This feature can be characterized by throttling stiffness, which reflects the ability to resist the load change of the notch to maintain flow stability. It is defined as the first derivative of pressure drop with respect to flow rate. The throttling stiffness (T) can be derived from the orifice flow formula (Eq. (2)) as:
T¼
pffiffiffiffiffiffi 2q 0:5 dDp ¼ Dp dQ Cd A
pffiffiffiffiffiffi Dp0:5 2q KðXÞ
Acknowledgments The authors gratefully acknowledge the support provided by National Natural Science Foundation of China No. 50875122, Jiangsu Province Science and Technology Support Program No. BE2011187 and Program Granted for Scientific Innovation Research of College Graduate in Jiangsu Province No. CXZZ13_0432.
ð16Þ References
From the analysis in section 2.2, CdA varies with spool opening X. Assuming CdA = K(X), the throttling stiffness T can be written as a function of X, Dp:
TðX; DpÞ ¼
1101
ð17Þ
By substituting the test data of pressure drop and opening of the three notches into Eq. (17), the values of the throttling stiffness are obtained and plot in Fig. 12 with X and Dp being the independent variables. Surface fitting is also carried out for visual analysis. It can be seen from Fig. 12 that the triangle-shape groove has the maximum stiffness under the same conditions, followed by the divergent U-shape groove and then the spheroid-shape groove. Nevertheless, they have almost the same variation trend with the openings and pressure drops. In each case, the greater the pressure drop Dp is, the greater the stiffness T is. In addition, a relatively high change rate of the stiffness appears when the pressure drop is small. As a result, in order to ensure that the notches possess sufficient stiffness, generally the minimum pressure drop is applied. When the inlet and outlet pressures of notch are the same, the stiffness decreases rapidly as the spool opening becomes larger. When the spool is close to fully opened status, the stiffness is near saturation and the pressure change will have weak impact on the stiffness. Among the three notches, the triangle-shape groove has the maximum stiffness at small openings, and most likely to have stable discharge.
5. Conclusions CFD simulation and experimental investigation of the flow characteristics of three different notches with their corresponding typical structural grooves are presented. The results show that the groove shape has significant effects on the flow area, the discharge characteristics, the jet flow angle, the steady flow force and the throttling stiffness of the spool valve. The study is starting from the calculations of the notch flow area based on the analysis of pressure distributions of the flow fields inside the notches. Different features reflected in the changes of the restricted locations inside the three notches along with the spool stroke are highlighted. The stable values of the discharge coefficients are derived from the quadratic polynomial fitting of the experimental data of pressure drop vs. flow rate. The transitional zones of the fluid flow through the notches range from 10 to 30 in terms of Re0.5, which show up earlier than the classic pipe flow. The steady flow forces with respect to the opening are calculated and are in good agreement with the experimental results except for some small differences. Among the three types, the triangle-shape groove has the maximum throttling stiffness, followed by the divergent U-shape groove and then the spheroid-shape groove. The above results demonstrate that the flow characteristics of the spool valve depend strongly on the groove shape and provide effective guidance for the design of hydraulic valve.
[1] Lisowski E, Rajda J. CFD analysis of pressure loss during flow by hydraulic directional control valve constructed from logic valves. Energy Convers Manage 2013;65:285–91. http://dx.doi.org/10.1016/j.enconman.2012.08.015. [2] Lisowski E, Czyzycki W, Rajda J. Three dimensional CFD analysis and experimental test of flow force acting on the spool of solenoid operated directional control valve. Energy Convers Manage 2013;70:220–9. http:// dx.doi.org/10.1016/j.enconman.2013.02.016. [3] Pan XD, Wang GL, Lu ZS. Flow field simulation and a flow model of servo-valve spool valve orifice. Energy Convers Manage 2011;52:3249–56. http:// dx.doi.org/10.1016/j.enconman.2011.05.010. [4] Posa A, Oresta P, Lippolis A. Analysis of a directional hydraulic valve by a Direct Numerical Simulation using an immersed-boundary method. Energy Convers Manage 2013;65:497–506. http://dx.doi.org/10.1016/ j.enconman.2012.07.012. [5] Amirante R, Del Vescovo G, Lippolis A. Evaluation of the flow forces on an open centre directional control valve by means of a computational fluid dynamic analysis. Energy Convers Manage 2006;47:1748–60. http://dx.doi.org/ 10.1016/j.enconman.2005.10.005. [6] Yuan QH, Li PY. Using steady flow force for unstable valve design: modeling and experiments. ASME J Dyn Syst Meas Control 2005;127:451–62. [7] Viall EN, Zhang Q. Determining the discharge coefficient of a spool valve. In: Proceedings of the American control conference, Chicago; June 2000. p. 3600– 04. [8] Ji H, Fu X, Yang HY. Measurement on pressure distribution of non-circular opening spool valve. Chin J Mech Eng 2004;40(4):99–102. [9] Ji H, Wang DS, Ryu SH, Fu X. Flow control characteristic of the orifice in spool valve with notches. Trans Chin Soc Agricult Mach 2009;40(1):198–202. [10] Ji H, Fu X, Yang HY. Study on steady flow force of non-circular opening spool valve. Chin J Mech Eng 2003;39(6):13–7. [11] Ryu SH, Ochiai M, Ueno K, Fu X, Ji H. Analysis of flow force in valve with notches on spool. In: Proceedings of the sixth international conference on fluid power transmission and control, Tallinn; September 2005. p. 435–38. [12] He XH, Huang GQ, Yang YS, Li ZY. Experimental research on the flow characteristics of ball valve orifice. Chin J Mech Eng 2004;40(8):30–3. [13] Ye QF, Chen JP. Dynamic analysis of a pilot-operated two-stage solenoid valve used in pneumatic system. Simul Model Pract Theory 2009;17:794–816. [14] Chattopadhyay H, Kundu A, Saha BK, Gangopadhyay T. Analysis of flow structure inside a spool type pressure regulating valve. Energy Convers Manage 2012;53:196–204. http://dx.doi.org/10.1016/ j.enconman.2011.08.021. [15] Chern MJ, Wang CC, Ma Ch. Performance test and flow visualization of ball valve. Energy Convers Manage 2007;31:505–12. http://dx.doi.org/10.1016/ j.expthermflusci.2006.04.019. [16] Amirante R, Moscatelli PG, Catalano LA. Evaluation of the flow forces on a direct (single stage) proportional valve by means of a computational fluid dynamic analysis. Energy Convers Manage 2007;48:942–53. http://dx.doi.org/ 10.1016/j.enconman.2006.08.024. [17] Borghi M, Milani M, Paoluzzi R. Stationary axial flow force analysis on compensated spool valve. Int J Fluid Power 2000;1:17–25. [18] Yang YS, Semini C, Tsagarakis NG, Caldwell DG, Zhu YQ. Water hydraulics – a novel design of spool-type valves for enhanced dynamic performance. In: Proceeding of the 2008 IEEE/ASME international conference on advanced intelligent mechatronics, Xi’an; July 2008. p. 1308–14. [19] Miller R, Fujii Y, McCallum J, Strumolo G, Tobler W, Pritts C. CFD simulation of steady-state flow forces on spool-type hydraulic valves. SAE Technical Paper Series; 1999. p. 295–307. [20] Krishnaswamy K, Li PY. On using unstable electrohydraulic valves for control. ASME J Dyn Syst Meas Control 2002;124:183–90. [21] Du XW, Zou J, Fu X, Ji H, Yang HY. Effect of throttling grooves structure on cavitation noise. J Zhejiang Univ (Eng Sci) 2007;41(3):456–65. [22] Fang WM, Cheng LL, Fu X, Zhou H, Yang HY. Investigation on steady-state flow force of spool valve with U-grooves. J Zhejiang Univ (Eng Sci) 2010;44(3):576–80. [23] Borghi M, Milani M, Paoluzzi R. Influence of notches shape and number of notches on the metering characteristics of hydraulic spool valves. Int J Fluid Power 2005;3:5–18. [24] Chen JS, Liu XH, Yuan WR, Wang TJ, Zhao F. Dynamic characteristics of typical hydraulic notches. J Southwest Jiaotong Univ 2012;47(2):325–32. [25] Zhou S, Xu B, Yang HY. Flow force compensation of high speed on/off valve. Chin J Mech Eng 2006;42:5–8.
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Effects of groove shape of notch on the flow characteristics of spool valve Yi Ye, Chen-Bo Yin ⇑, Xing-Dong Li, Wei-jin Zhou, Feng-feng Yuan Institute of Automobile and Construction Machinery, Nanjing Tech University, Nanjing 211816, China
a r t i c l e
i n f o
Article history: Received 23 February 2014 Accepted 26 June 2014 Available online 15 July 2014 Keywords: Flow characteristics Spool valve Notch Groove shape Computational fluid dynamics (CFD)
a b s t r a c t The grooves of notches of hydraulic spool valves are usually designed into various shapes for their desired flow characteristics. The aim of this paper is to clarify the effects of the groove shape on the flow characteristics through computational fluid dynamics (CFD) and experimental investigations. The RNG k–e turbulence model is used to simulate the pressure distributions of the flow fields inside three notches with their corresponding typical structural grooves in order to analyze the changes of restricted locations along with the openings and, furthermore, to calculate the flow areas of the notches. The accuracy of the employed model is demonstrated by comparing the computational results with the experimental data. Additionally, the flow rate vs. pressure drop data obtained from the experiment is fitted by least square method. On this basis, the discharge coefficient as a function of groove geometry, flow condition, fitting coefficients and its stable value is deduced, proving to be quite consistent with the experimental result. Thanks to the jet flow angles estimated by CFD simulation, the steady flow forces are calculated, which show good agreement with the experimental results except for some small differences. Finally, the throttling stiffness of the three notches is investigated, with that of divergent U-shape groove falls between spheroid-shape groove and triangle-shape groove. Similar results are found for steady flow force. The results indicate that the groove shape has significant effects on the flow characteristics (flow area, discharge characteristic, jet flow angle, steady flow force and throttling stiffness) of spool valve. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Work performance of a hydraulic system is largely determined by the hydraulic valve. And fluid flow control of the hydraulic valve essentially depends on the throttling form of notches. Notches with throttling grooves are widely used in high performance hydraulic valves where excellent precision and stability are required due to the following advantages, such as diverse structure, wide range of flow rate, the gradient of flow area being easy to control, good stability in the case of small flow, abundant flow control characteristics. Some scholars have launched a number of pioneering researches to study the relationship between the valve geometry and its flow characteristics. Lisowski et al. [1] proposed a logic type directional control valve with reduction of pressure losses over 35%, while the difference taken from CFD analysis and experimental research does not exceed 5%. They [2] also proposed the calculation of flow forces (pressure force and viscous force) acting on the spool of solenoid operated directional control valve using 3D CFD modeling along with k–e turbulence model. Pan et al. [3] analyzed the discharge
⇑ Corresponding author. Tel./fax: +86 025 58139599. E-mail address: [email protected] (C.-B. Yin). http://dx.doi.org/10.1016/j.enconman.2014.06.081 0196-8904/Ó 2014 Elsevier Ltd. All rights reserved.
characteristics of servo-valve spool valve under both laminar and turbulent flow by the CFD method and derived a formula for discharge coefficient and Reynolds number. Posa et al. [4] carried out an analysis of the discharge coefficient and the flow force of a directional valve utilizing 2D CFD method and, in addition, the fluid-body interaction had been represented by an immersedboundary technique. Amirante et al. [5] performed a complete 3D numerical simulation to deal with the driving forces acting on an open center directional control valve. Yuan and Li [6] highlighted the important effects of two often ignored components, viscosity effect and non-metering momentum flux, on the steady flow force for the purpose of achieving outstanding dynamics performances. All the above researches have demonstrated that the CFD method has advantages to predict flow characteristics of valve and is efficient in capturing many partial details in flow field. Moreover, an experimental procedure had been proposed to determine the discharge coefficient of a spool valve as a function of opening and pressure drop in Ref. [7]. Ji et al. [8] developed a pressure distribution and noise measurement test equipment of hydraulic valve with a mobile valve sleeve, and discussed the interior relations among the throttling groove structure, pressure distribution and noise characteristics. Furthermore, relevant works with reference to flow area, flow coefficient and steady flow force of spool valve
1092
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
with notches were experimentally investigated in other studies of Ji et al. [9–11]. Some other notable works aimed at flow characteristics include studies on flow test of ball valve by He et al. [12], dynamic analysis of a two-stage solenoid valve flow by Ye et al. [13], flow structure inside a spool type pressure regulating valve by Chattopadhyay et al. [14] and visualization experiment of flow patterns and cavitation phenomena by Chern et al. [15]. In general, these researches were focused on global performances of valves. Unfortunately, the influences of valve geometry on flow characteristics were not highlighted. Some investigations proposed functional structures achieved through modifying or changing the geometry of valve to improve flow force balance, anti-cavitation ability, flow capacity, etc. Amirante et al. [16] numerically studied the effect of optimized groove and compensation profile on reducing the flow force of a proportional valve, and also provided the corresponding experimental verification. Borghi et al. [17] presented three different valve port connections which were used to provide compensation to stationary axial flow forces of a reference spool valve, and the compensation effect was verified by experiment. Many other innovative designs were also presented, for instance, arc notches in spool land and sleeve [18], additional parallel and compensatory channels in a solenoid operated directional control valve [2], tapered structure on spool rod [19] and rotated channel in non-metering port [20]. And positive effects of the novel structures in the above articles were also shown. However, limited work [21–23] has been carried out to study the influence of groove shape on flow characteristics although groove shape, to a great extent, may determine the observable behavior of valve. Moreover, the effects of groove shape on flow characteristics still lack systematic research. By investigating the effects of groove shape on flow characteristics of spool valve from multiple perspectives, this study fills up the gap of previous studies by proposing three types of throttling grooves with different structural features, namely spheroid-shape groove, triangle-shape groove and divergent U-shape groove. Following a brief description of groove structures, the pressure distributions of the flow fields inside the notches are numerically simulated so as to analyze the changes of restricted locations along with the increasing spool opening and to calculate the equivalent notch area. Then, a least square method is utilized to fit the test data of flow rate vs. pressure drop. The discharge coefficient as a function of groove geometry, flow condition, fitting coefficients and its stable value is deduced, and stable values of the discharge coefficient are obtained and discussed. After that, the steady flow force is calculated as the jet flow angle is estimated by CFD numerical simulation. Finally, the throttling stiffness of the notch is also investigated. The results provide effective guidance for the design of high-performance hydraulic valve. 2. CFD analysis 2.1. Groove structures The three types of throttling grooves with typical structural features are shown in Fig. 1. Fig. 1a is a spheroid-shape groove and Fig. 1b is a triangle-shape groove. Fig. 1c is a divergent U-shape groove with a relatively complex structure, which could be considered as being overlaid in parallel by a U-shape part and a divergent one. In Fig. 1, X is the spool opening; A1 and A2 are the axial and radial cross-sections respectively, and both cross the throttling edge. A3 is the cross-section which crosses both the throttling edge and the lowest point of the groove. Amin is the smallest cross-section across the throttling edge and which is usually perpendicular to the bottom. To demonstrate the effects of the structural features on the flow characteristics, the lengths of the throttling grooves in
the axial direction are set to be equal (3 mm). To reduce the impact of radial imbalance force on the experimental results, two throttling grooves are symmetrically processed in the circumferential direction on the spool. 2.2. Modeling and simulation Control of fluid flow by notches is accomplished through regulating the opening so as to change the flow area. Therefore, determining the relationship between the opening and flow area is the foundation of carrying out the flow characteristics investigation. Generally, several cross-sections in the notch contribute to the throttling effect, resulting in complex flow structure and pressure distribution of the fluid field inside the notch. For single-stage throttling groove in the previous studies [9,10,23,24], throttling effect was usually attributed to axial, radial or minimum cross-section across throttling edge (defined as throttling cross-section thereof). But in fact, as the opening changes, the throttling crosssection varies dynamically. According to the Bernoulli law, flow velocity will rise sharply along with a rapid pressure drop when fluid flow through the throttling cross-section. In other words, a rapid pressure drop only happens in the throttling cross-section of the notch. Thus, in order to obtain the accurate location of the throttling cross-section, CFD numerical simulation method is applied to analyze the pressure distribution inside the notch. The grid model is established using half part of the flow field inside the valve for it is an axis symmetrical structure. As shown in Fig. 2 (take spheroid-shape groove in X = 0.6 mm as an example), it is divided by the application of tetrahedral mesh which contains about 450,000 cells. Grid models for triangle-shape groove and divergent U-shape groove cases contain about 435,000 and 455,000 cells, respectively. What’s more, it can be seen from Fig. 2 that the grid gradually becomes intensive in the process of getting close to the wall and the inlet and outlet boundaries of the notch. The RNG k–e turbulence model is selected, and boundary conditions of inlet and outlet pressures are imposed. In addition, solution accuracy is set to be 105, and hydraulic oil with viscosity of 46 cSt and density of 896 kg/m3 is chosen as the medium. Gridindependent test of the model is carried out with various grid sizes to make sure the obtained numerical solutions are independent on grid density. Five positions (X = 0.6, 1.4, 2.0, 2.8, 3.1 mm) during opening process are selected to be analyzed. 2.3. Pressure distribution The pressure distributions in the symmetrical surface of the notches at the corresponding positions are shown in Fig. 3. The pressure drop through the spheroid-shape groove is concentrated on cross-sections A2 and A3 within the entire range of the spool stroke. In other words, the throttling effect of the spheroid-shape groove is provided by cross-sections A2 and A3 together, showing two-level throttling characteristics. In particular, with the increase of the opening, the proportion of the pressure drop changes gradually from cross-section A2 to A3. The triangle-shape groove is a typical divergent structure, whose pressure distribution characteristics are relatively simple. In the triangle-shape groove, cross-sections A1 and A2 do not have throttling function. The pressure drop is consistently centralized in cross-section Amin. The divergent U-shape groove has a more complex behavior of the pressure distribution than the other two types. Migration of throttling cross-sections exists in this type as shown in Fig. 3. At small openings, pressure drop is consistently centralized in crosssections A1 and A2. In addition, even though cross-section A2 bears more pressure drop, two-level throttling characteristics dominate in the pressure distribution. With the increasing opening, the
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
1093
Fig. 1. Geometric structures of: (a) the spheroid-shape groove; (b) the triangle-shape groove and (c) the divergent U-shape groove.
cross-section A2 and inclined cross-section A3 caused by simplified treatment are both verified to be less than 4%, demonstrating its feasibility. MATLAB is used to develop the flow area calculation procedure of the notch. When the flow area curves are drawn, A/ Amax and X/Xmax are respectively set as ordinate and abscissa as shown in Fig. 4. In addition, A1, A2, A3 are the area sizes of A1, A2, A3, respectively. As shown in Fig. 4a, Aeq is the minimum and should be regarded as the notch flow area value of the spheroid-shape groove during the entire opening process. In the first half of the spool stroke, Aeq is almost equal to A2 for A2 is much smaller than A3. However, A2 and A3 contribute more equally to Aeq during the second half of the spool stroke as the ratio of A3 to A2 gradually decreases to 1. For the triangle-shape groove (Fig. 4b), the ratio of A1 to A2 remains constant. In this case, the notch flow area is set to the value of Aeq, which is also equal to Amin. With regard to the divergent Ushape groove, there is an intersection point in the spool opening at X = 2.0 mm which is approximately 0.667 in term of X/Xmax from Fig. 4c. When X is smaller than 2.0 mm, the notch flow area equals Aeq. When X is larger than 2.0 mm, it changes to the minimum throttling cross-sectional area Amin. Fig. 2. Grid model for the spheroid-shape groove case.
throttling effect of cross-section A2 declines gradually. Meanwhile, the pressure loss at cross-section A1 migrates to cross-section Amin. The migration process is still characterized by the two-level throttling. Later in the remaining opening process, as the area of crosssection A2 continually increases, the throttling effect of cross-section A2 decreases until it disappears. At the same time, single-level throttling feature appears and Amin becomes the centralized crosssection for the total pressure drop.
2.4. Flow area analysis Currently, in the study of flow area of the notch with throttling groove, an effective approach is applied by replacing multiple throttling cross-sections with an equivalent throttling cross-section. In present paper, the minimum value between Amin and Aeq is taken as the notch flow area where Amin and Aeq are the area sizes of Amin and the equivalent throttling cross-section, respectively. Furthermore, Aeq is set to the value of the equivalent area of A2 and A3 in series in the spheroid-shape groove and A1 and A2 in series in the other two types according to the analysis of pressure distribution in section 2.3. In fact, the top surface of the throttling groove is a circular surface. The boundary lines of the enclosed surface are space curves. Therefore, obtaining the specific analytical solution of the flow area by applying numerical integration method or solving nonlinear function becomes really complicated [24]. As such, the circular surface is simplified to a plane when the flow area is calculated. While the spheroid-shape groove has the maximum circumferential span among the three throttling grooves, the area errors of its radial
3. Experimental setup A sketch of the notch flow characteristics experimental setup is presented in Fig. 5. The inlet and outlet pressures of the notch are respectively adjusted by a variable pump (range 0–150 l/min, class 0.2 absolute) and a counterbalance valve. The maximum pressure of the system is set to be 30 MPa by a safety valve. The mechanical oscillation of the testing apparatus is effectively suppressed by the hose and filter, and so is the interference caused by the pulsation of the fluid flow. The movement of the valve spool is adjusted by a ball screw which is driven by a stepping motor. Two pressure sensors (range 0–60 MPa, class 0.2 absolute) are arranged on the valve chamber. The rate of flow through the tested notch is measured by a flow meter (range 0–150 l/min, class 0.1 absolute). The axial force acting on the spool is measured by a force sensor (range 0–150 N, class 0.1 absolute). A compensation method [25] is used to test the steady flow force. Also, hydraulic oil with viscosity of 46 cSt and density of 896 kg/m3 is selected as the working fluid in the test. Considering that the test result is greatly influenced by the form error of the spool structure (such as the circular bead of the groove) when the spool opening is at the limits, the opening range of 0.4 mm to 2.8 mm is set as the scope of study. What’s more, three feature positions of X = 0.6, 1.4, 2.2 mm are selected to be emphatically analyzed within the range. 4. Results and discussion 4.1. Discharge characteristics MATLAB is used to deal with the obtained data from the test. The curves of the flow rate vs. pressure drop when the openings
1094
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
Fig. 3. Pressure distributions inside the notches at positions: (a) X = 0.6 mm; (b) X = 1.4 mm; (c) X = 2.0 mm; (d) X = 2.8 mm and (e) X = 3.1 mm.
of the three notches are at the feature positions are shown in Fig. 6. It is worth noting that the scale of the horizontal axis of the flow rate is different in different plots. From Fig. 6, it is obvious that the spheroid-shape groove has the strongest flow capacity,
followed by the divergent U-shape groove and then the triangleshape groove under the same pressure drop. In addition, the corresponding curves of CFD simulation results are also provided in Fig. 6. A comparison between the experimental and the CFD
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
1095
Fig. 5. Sketch of the experimental setup: (1) variable pump; (2) counterbalance valve; (3) safety valve; (4) hose; (5) filter; (6) stepping motor; (7) ball screw; (8) valve spool; (9) pressure sensor; (10) flow meter; (11) slid platform and (12) pedestal.
Dp ¼ a1 Q 2 þ a2 Q
ð1Þ
The values of coefficients a1, a2 are presented in Table 1. All the values of the fitting regression coefficients R are higher than 0.994, indicating the fitted curves of the flow rate vs. pressure drop are in good agreement with the experimental data (Fig. 6). The fluid flowing through the throttling groove can be described by the orifice flow formula:
Q ¼ Cd A
sffiffiffiffiffiffiffiffiffi 2Dp
ð2Þ
q
where Cd is the discharge coefficient, A is the notch flow area, q is the density of the fluid. As shown in Eq. (2), the discharge characteristic of the notch is determined by parameters Cd, A, q and Dp. Cd and A are inherent properties of the notch, which are only related to the throttling groove shape. The notch flow area A reflects the flow capacity of the notch. Greater A means stronger flow capacity. Cd reflects the resistance characteristic of the notch. The greater Cd is, the smaller the flow resistance of the notch is. When the pressure difference between the inlet and outlet of the flow field is certain, the rate of flow through the notch becomes larger as A and Cd increase. Eq. (2) can be rearranged as
Dp q ¼ Q 2 2C 2d A2
ð3Þ
And likewise, the ratio of Dp to Q2 can be expressed from Eq. (1) as follows
Dp a2 ¼ a1 þ Q Q2 Fig. 4. Plot of the flow area vs. opening for: (a) the spheroid-shape groove; (b) the triangle-shape groove and (c) the divergent U-shape groove.
simulation results clearly indicates that, in each case the dependence of the flow rate on the pressure drop is not exactly the same yet very close. The differences between the test results and their corresponding simulation values do not exceed 7%. The distribution of the flow rate vs. pressure drop data conforms to a quadratic curve with its vertex being exactly the original point. Therefore, curve fitting of the data obtained from the experiment is conducted by the least square method. The fitting relation of the pressure drop Dp and the flow rate Q can be expressed as:
ð4Þ
Since the fluid medium is selected and the groove shape as well as the opening is certain, the corresponding parameters q, A can be determined. And a1, a2 are known from Table 1. Combining Eqs. (3) and (4), the discharge coefficient Cd as a function of Q can be expressed as
Cd ¼
1 A
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
2ða1 þ a2 =Q Þ
ð5Þ
When increasing flow rate Q continually to infinity, the discharge coefficient will gradually approach a stable value (Cdst):
C dst ¼ lim C d ¼ Q!1
1 A
rffiffiffiffiffiffiffiffi
q
2a1
ð6Þ
1096
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101 Table 1 Polynomial coefficients for data interpolation. Notch structure
Opening (mm)
a1 (MPa/(l/ min)2)
a2 (MPa/(l/ min))
Spheroid-shape groove
0.6 1.4 2.2
2.573e2 2.647e3 9.170e4
7.299e03 7.397e04 1.700e03
Triangle-shape groove
0.6 1.4 2.2
2.083e + 1 7.605e1 1.347e1
8.606e01 9.459e02 2.654e02
Divergent U-shape groove
0.6 1.4 2.2
1.168e1 2.103e2 1.058e2
3.962e03 5.121e03 2.785e03
Table 2 Stable value of the discharge coefficient. Notch structure
Opening (mm)
Cdst
Spheroid-shape groove
0.6 1.4 2.2
0.747 0.682 0.620
Triangle-shape groove
0.6 1.4 2.2
0.720 0.692 0.666
Divergent U-shape groove
0.6 1.4 2.2
0.651 0.646 0.652
The relation between the discharge coefficient and the flow rate is described by Eq. (7). However, it cannot tell the change of the discharge coefficient with the flow condition. Therefore, Reynolds number (Re) of the fluid flowing through the notch is introduced to characterize the flow condition and defined as:
Re ¼
mqDh l
ð8Þ
where m is the flow velocity, which can be expressed as m = Q/2Amin, and l is the dynamic viscosity of the hydraulic oil. Dh is the hydraulic diameter, which is defined as four times the ratio of wet crosssectional area to wet perimeter length. In this study, the flow regime of the flow field inside the notch is determined by the maximum flow velocity which occurs in the cross-section with the minimum area. Therefore, to calculate the Reynolds number the values of the wet cross-sectional area and wet perimeter length are got in cross-section Amin. Thus, Dh is expressed as
Dh ¼
4Amin C
ð9Þ
where C is the perimeter length of Amin. Combining Eqs. (8) and (9), the relation between the flow rate and the Reynolds number can be obtained: Fig. 6. Curves of the flow rate vs. pressure drop of: (a) the spheroid-shape groove; (b) the triangle-shape groove and (c) the divergent U-shape groove.
Q¼
lCRe 2q
ð10Þ
Substitute Eq. (10) into Eq. (7), and after arranging we have
In this case, the fluid flow through the notch is fully turbulent. The stable values of the discharge coefficient at the feature positions of the three notches are calculated using Eq. (6) and the results are shown in Table 2. Put Cdst into Eq. (5), it becomes:
1 Cd ¼ A
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ C dst 1 þ a2 =a1 Q 2a1 ð1 þ a2 =a1 QÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
ð7Þ
C d ¼ C dst
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 2qa2 =a1 lCRe
ð11Þ
Therefore, the discharge coefficient is a function of groove geometry, flow condition, fitting coefficients and its stable value. Substitute the values of Cdst, a1, a2, l, q and C into Eq. (11), plots of the dependences of the discharge coefficients on the Reynolds number are obtained for different notches at the feature openings as shown in Fig. 7. Re0.5 is set as the abscissa and its scale is different in different plots. After data processing of the test results using
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
1097
orifice flow formula directly, we can see that the calculated results are in good agreement with the experimental ones. It is worth noting that in all samples the discharge coefficients first increase rapidly with regard to Reynolds number in the laminar flow region, and then gradually achieve the stable values around the transitional zones which range from 10 to 30 in terms of Re0.5. As the throttling groove is narrow, the fluid flowing through the notch is quite fast and seriously constrained by its wall, leading to a constant change of flow direction. In this case, laminar flow is easy to be damaged as viscous forces between liquid particles weaken. Consequently, inertia force of the liquid plays a dominant role, resulting in a transitional zone in advance. With the increasing opening, the discharge coefficient of the spheroid-shape groove reaches the stable value Cdst more laggingly, while the triangleshape groove and the divergent U-shape groove show no obvious change. The relation between the stable value of the discharge coefficient and the opening is significant for designing hydraulic valves with diverse flow control functions. Within the opening range of 0.4–2.8 mm, the discharge characteristic test is performed every 0.2 mm. The pressure difference between the inlet and the outlet of the notch is set large enough to obtain the stable value of the discharge coefficient in each test. Substitute the test data and the notch flow area into Eq. (2), we obtain the discharge coefficient distribution with the opening as shown in Fig. 8. From the trend of the curve, the impact of the opening on the discharge coefficient could be summarized as follows: (1) The discharge coefficient of the spheroid-shape groove has an approximately negative linear relationship with the spool opening. At small openings, A3 A2, radial cross-section A2 plays the main throttling role. And the flow resistance of the throttling groove is small, meaning a relatively large Cd. With the increase of the opening, A2 increases rapidly. The area ratio of A3/A2 gradually decreases to 1 as the twolevel throttling feature becomes obvious and Cd decreases continuously. (2) When the spool opening X < 0.8 mm, the discharge coefficient of the triangle-shape groove increases quickly from 0.639 to 0.71 as a result of clogging which often occurs in fluid flow through the notch under the condition of small flow area and large pressure difference. On the contrary, it keeps decreasing when X > 1 mm. This phenomenon is mainly attributed to the occurrence of cavitation in the partial low-pressure zones of the throttling cross-section, which has been observed inside a similar notch with V-shape groove by visualization experiment [21]. The occurrence of cavitation decreases the actual flow area of the notch. The influence becomes more significant as the opening increases under a constant pressure drop which leads to a continuous decline of the discharge coefficient. (3) As for the divergent U-shape groove, when X < 1 mm, the discharge coefficient drops quickly from 0.683 to 0.587. As X > 1 mm, the discharge coefficient increases gradually. When X = 1.8 mm, Cd increases to 0.648 and remains almost unchanged thereafter. According to the flow area characteristics, in the initial opening process, the pressure drop is mostly concentrated in the cross-section A2. And the flow resistance of the throttling groove is mainly caused by A2, showing single-level throttling characteristic. With the increase of the opening, A2 increases drastically. At approximately X = 1 mm, the total pressure drop distributes equally in the cross-sections A1 and A2 as A1/A2 approaches 1, resulting in the maximum throttling effect during the entire opening process. As a consequence, Cd decreases to its lowest
Fig. 7. Plots of Cd vs. Re0.5 for: (a) the spheroid-shape groove; (b) the triangle-shape groove and (c) the divergent U-shape groove.
value. Thereafter, A2 continues to increase drastically. As the difference of A1 and A2 becomes larger, the two-level throttling characteristic weakens. When the opening exceeds 2 mm, the throttling effect is solely caused by Amin, showing single-level throttling characteristic.
1098
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
Fig. 9. Schematic diagram of fluid flow through the notch.
Fig. 8. Discharge coefficient distribution with the opening.
Based on the above analysis, we can see that the three throttling grooves can be applied to the hydraulic system for different purposes according to their corresponding discharge characteristics. At small openings, the discharge coefficient of the spheroid-shape groove is great, which is good at quickly building up the working pressure for the applied hydraulic system. The discharge coefficient of the triangle-shape groove decreases gently in relation to the opening after its initial rise, which is beneficial to fine tuning of the flow rate and smooth running of the hydraulic system. As to the divergent U-shape groove, the discharge coefficient is big when the opening is small and followed by a rapid decline, which is advantage to avoiding the vibration occurred during startup process of the hydraulic system. 4.2. Steady flow force Flow force is one of the key factors that determine the pros or cons of the hydraulic control valve performance. It not only affects the operating force on the spool, but also causes the self-excited vibration of the components of the hydraulic system. Accurate calculation of the flow force is the prerequisite for control accuracy and force balance of the spool. As shown in Fig. 9, the control volume under flow force is the region surrounded by spool wall, notch horizontal surface and valve body underside. According to the momentum conservation law, the axial force balance equation for the control volume can be expressed in vector form (positive to the right) as follows:
~ F land þ ~ F body þ ~ F rod þ qQ ð~ v1 ~ v2Þ ¼ 0
As the area of the outlet is considerably larger than that of the inlet, the outlet velocity v2 is much smaller than the inlet velocity v1. Furthermore, the downstream flow is constrained by the spool wall with the outlet flow angle h2 being close to 90°. As a consequence, the outlet efflux force, namely the second term on the right side of Eq. (14) can be ignored. According to the literature [18,20], the viscous force can be neglected because it is too small comparing with the inlet efflux force (the first term on the right side of Eq. (14)), i.e., Fbody = 0. It’s worth noting that the inlet velocity v1 is not the jet flow velocity at the notch, but the velocity on the inlet surface (i.e. A2) of the control volume. By substituting Eq. (2) into Eq. (14), steady flow force can be further expressed as:
Ff ¼
2C 2d A2 cot h1 Dp A2
ð15Þ
Table 3 Jet flow angles at variable openings. Opening (mm)
Jet flow angle (°) Spheroid-shape groove
Triangle-shape groove
Divergent U-shape groove
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
78.5 74.8 72.0 70.3 69.1 67.9 67.2 66.6 66.2 65.7 65.3 64.9 64.6
44.6 45.2 44.2 43.9 43.8 42.5 43.4 44.4 44.5 44.2 43.9 44.5 43.6
67.0 66.4 66.1 65.8 65.3 64.9 64.1 63.4 63.0 62.7 62.5 62.2 61.1
ð12Þ
~ F land is pressure force vector subjected by the spool lands; ~ F body is F rod is viscous viscous force vector subjected by the valve body; ~ force vector subjected by the spool rod; ~ v 1 and ~ v 2 are inlet velocity F body vector and outlet velocity vector, respectively. The direction of ~ F rod are toward the left as the fluid flow to the right, which can and ~ be judged easily from Fig. 9. F f that the spool experiences from fluid The steady flow force ~ flow is equal to the sum of pressure forces ~ F land0 acting on the spool lands and viscous force ~ F rod0 acting on the spool rod, which is given in vector form as:
~ Ff ¼ ~ F land0 þ ~ F rod0
ð13Þ
~ F land and ~ F land0 are equal but in the opposite direction, the same as ~ F rod0 . The above two equations lead to the following scalar F rod and ~ form:
F f ¼ qQ v 1 cos h1 qQ v 2 cosh2 F body
ð14Þ
Fig. 10. Steady flow force at variable openings.
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
When the opening is certain, Cd, A, A2, h1 are fixed values. Assuming coefficient Ks = 2Cd2A2coth1/A2 is constant, the steady flow force Ff is proportional to pressure drop Dp. Flow structure
1099
in the control volume cannot be observed in the experiment, so CFD simulation is used to determine the jet flow angle h1. Table 3 summarizes the jet flow angles of the three throttling grooves
Fig. 11. Contours of pressure and velocity distributions of the divergent U-shape groove at opening: (a) X = 0.6 mm; (b) X = 1.4 mm and (c) X = 2.2 mm.
1100
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
at different openings. The jet flow angle variation range of the spheroid-shape groove reaches 14°, with great changes at the initial openings. In the intermediate positions, it is very close to the theoretical value of 69° derived by Von Mises. The jet flow angle of the triangle-shape groove is significantly affected by the flowdirection guide of the bottom slope and independent of the opening. Throughout the opening, the jet flow angle is small, and maintains at about 44°. The jet flow angle of the divergent U-shape groove decreases roughly linearly from 67° to 61.1°. It is noteworthy that for the steady flow force experiment, the pressure difference between the inlet and the outlet is set to 20 MPa for each test, with the error range of ±0.5 MPa, and the same pressure difference is applied for the corresponding calculation. Fig. 10 shows the comparison between the calculated and experimental values of the steady flow force at different openings. It can be seen that theoretical calculations can fully reflect variation trends of the experimental steady flow forces. The steady flow forces of the triangle-shape groove and the divergent U-shape groove increase roughly linearly with the increasing opening. At very small openings, the steady flow force of the triangle-shape groove appears to be negative, tending to open the valve. This appearance may be because of the occurrence of complex vortex and reflux in the exit section of the control volume, which makes the axial exit momentum non-ignorable or even larger compared to the axial entry momentum. At the same time the axial entry momentum is very tiny as the discharge coefficient and the notch flow area are both at a low level. Similar phenomenon was also observed in the experimental investigation of spool valve with Ushape grooves at small openings [22]. At the same opening, the test values are slightly smaller than the calculated values for all the three cases. The differences gradually increase with the increasing openings. A hypothesis has been made for the calculation of the steady flow force that the pressure difference between the inlet and the outlet of the flow field is considered wholly concentrated on the throttling section. This hypothesis actually presents a larger pressure drop through the throttling section in the calculation than the test, leading to the differences between the calculated values and the test values. Fig. 11 provides the velocity and pressure distributions in the symmetrical plane at opening X = 0.6, 1.4, 2.2 mm of the divergent U-shape groove under the same pressure drop of 5 MPa. At X = 0.6 mm, the flow velocity attains its maximum value inside the groove region, which means the pressure drop is mostly restricted to the throttling section as shown in Fig. 11a. With the increase of the opening, the region of the maximum velocity migrates downstream of the control volume. Correspondingly, the pressure losses dissipated at non-throttling sections become more and more significant with reference to the main pressure drop through the throttling groove as can be seen in Fig. 11b and c. Similar results have been obtained for the other two types (data not shown). In addition, with the increase of the opening, the outlet area reduces and the outlet flow angle gradually deviates from 90°due to diversion by the spool shoulder, both causing a slight increase of the outlet efflux force. For these reasons, the differences between the calculated values and the corresponding test values of the steady flow forces gradually enlarge with the increasing opening. The same reasons apply to the spheroid-shape groove as well as the triangle-shape groove. When the notch is close to fully opened status, the flow resistance of the throttling section virtually disappears. And the test values of the steady flow force increase suddenly while the flow rate increases rapidly (Fig. 10). 4.3. Throttling stiffness When the discharge of the notch extremely depends on the external load, minor change of pressure difference will still cause
Fig. 12. Plot of throttling stiffness with variable pressure drops and openings for: (a) the spheroid-shape groove; (b) the triangle-shape groove and (c) the divergent U-shape groove.
Y. Ye et al. / Energy Conversion and Management 86 (2014) 1091–1101
severe discharge oscillation, leading to instability of the flow control. This feature can be characterized by throttling stiffness, which reflects the ability to resist the load change of the notch to maintain flow stability. It is defined as the first derivative of pressure drop with respect to flow rate. The throttling stiffness (T) can be derived from the orifice flow formula (Eq. (2)) as:
T¼
pffiffiffiffiffiffi 2q 0:5 dDp ¼ Dp dQ Cd A
pffiffiffiffiffiffi Dp0:5 2q KðXÞ
Acknowledgments The authors gratefully acknowledge the support provided by National Natural Science Foundation of China No. 50875122, Jiangsu Province Science and Technology Support Program No. BE2011187 and Program Granted for Scientific Innovation Research of College Graduate in Jiangsu Province No. CXZZ13_0432.
ð16Þ References
From the analysis in section 2.2, CdA varies with spool opening X. Assuming CdA = K(X), the throttling stiffness T can be written as a function of X, Dp:
TðX; DpÞ ¼
1101
ð17Þ
By substituting the test data of pressure drop and opening of the three notches into Eq. (17), the values of the throttling stiffness are obtained and plot in Fig. 12 with X and Dp being the independent variables. Surface fitting is also carried out for visual analysis. It can be seen from Fig. 12 that the triangle-shape groove has the maximum stiffness under the same conditions, followed by the divergent U-shape groove and then the spheroid-shape groove. Nevertheless, they have almost the same variation trend with the openings and pressure drops. In each case, the greater the pressure drop Dp is, the greater the stiffness T is. In addition, a relatively high change rate of the stiffness appears when the pressure drop is small. As a result, in order to ensure that the notches possess sufficient stiffness, generally the minimum pressure drop is applied. When the inlet and outlet pressures of notch are the same, the stiffness decreases rapidly as the spool opening becomes larger. When the spool is close to fully opened status, the stiffness is near saturation and the pressure change will have weak impact on the stiffness. Among the three notches, the triangle-shape groove has the maximum stiffness at small openings, and most likely to have stable discharge.
5. Conclusions CFD simulation and experimental investigation of the flow characteristics of three different notches with their corresponding typical structural grooves are presented. The results show that the groove shape has significant effects on the flow area, the discharge characteristics, the jet flow angle, the steady flow force and the throttling stiffness of the spool valve. The study is starting from the calculations of the notch flow area based on the analysis of pressure distributions of the flow fields inside the notches. Different features reflected in the changes of the restricted locations inside the three notches along with the spool stroke are highlighted. The stable values of the discharge coefficients are derived from the quadratic polynomial fitting of the experimental data of pressure drop vs. flow rate. The transitional zones of the fluid flow through the notches range from 10 to 30 in terms of Re0.5, which show up earlier than the classic pipe flow. The steady flow forces with respect to the opening are calculated and are in good agreement with the experimental results except for some small differences. Among the three types, the triangle-shape groove has the maximum throttling stiffness, followed by the divergent U-shape groove and then the spheroid-shape groove. The above results demonstrate that the flow characteristics of the spool valve depend strongly on the groove shape and provide effective guidance for the design of hydraulic valve.
[1] Lisowski E, Rajda J. CFD analysis of pressure loss during flow by hydraulic directional control valve constructed from logic valves. Energy Convers Manage 2013;65:285–91. http://dx.doi.org/10.1016/j.enconman.2012.08.015. [2] Lisowski E, Czyzycki W, Rajda J. Three dimensional CFD analysis and experimental test of flow force acting on the spool of solenoid operated directional control valve. Energy Convers Manage 2013;70:220–9. http:// dx.doi.org/10.1016/j.enconman.2013.02.016. [3] Pan XD, Wang GL, Lu ZS. Flow field simulation and a flow model of servo-valve spool valve orifice. Energy Convers Manage 2011;52:3249–56. http:// dx.doi.org/10.1016/j.enconman.2011.05.010. [4] Posa A, Oresta P, Lippolis A. Analysis of a directional hydraulic valve by a Direct Numerical Simulation using an immersed-boundary method. Energy Convers Manage 2013;65:497–506. http://dx.doi.org/10.1016/ j.enconman.2012.07.012. [5] Amirante R, Del Vescovo G, Lippolis A. Evaluation of the flow forces on an open centre directional control valve by means of a computational fluid dynamic analysis. Energy Convers Manage 2006;47:1748–60. http://dx.doi.org/ 10.1016/j.enconman.2005.10.005. [6] Yuan QH, Li PY. Using steady flow force for unstable valve design: modeling and experiments. ASME J Dyn Syst Meas Control 2005;127:451–62. [7] Viall EN, Zhang Q. Determining the discharge coefficient of a spool valve. In: Proceedings of the American control conference, Chicago; June 2000. p. 3600– 04. [8] Ji H, Fu X, Yang HY. Measurement on pressure distribution of non-circular opening spool valve. Chin J Mech Eng 2004;40(4):99–102. [9] Ji H, Wang DS, Ryu SH, Fu X. Flow control characteristic of the orifice in spool valve with notches. Trans Chin Soc Agricult Mach 2009;40(1):198–202. [10] Ji H, Fu X, Yang HY. Study on steady flow force of non-circular opening spool valve. Chin J Mech Eng 2003;39(6):13–7. [11] Ryu SH, Ochiai M, Ueno K, Fu X, Ji H. Analysis of flow force in valve with notches on spool. In: Proceedings of the sixth international conference on fluid power transmission and control, Tallinn; September 2005. p. 435–38. [12] He XH, Huang GQ, Yang YS, Li ZY. Experimental research on the flow characteristics of ball valve orifice. Chin J Mech Eng 2004;40(8):30–3. [13] Ye QF, Chen JP. Dynamic analysis of a pilot-operated two-stage solenoid valve used in pneumatic system. Simul Model Pract Theory 2009;17:794–816. [14] Chattopadhyay H, Kundu A, Saha BK, Gangopadhyay T. Analysis of flow structure inside a spool type pressure regulating valve. Energy Convers Manage 2012;53:196–204. http://dx.doi.org/10.1016/ j.enconman.2011.08.021. [15] Chern MJ, Wang CC, Ma Ch. Performance test and flow visualization of ball valve. Energy Convers Manage 2007;31:505–12. http://dx.doi.org/10.1016/ j.expthermflusci.2006.04.019. [16] Amirante R, Moscatelli PG, Catalano LA. Evaluation of the flow forces on a direct (single stage) proportional valve by means of a computational fluid dynamic analysis. Energy Convers Manage 2007;48:942–53. http://dx.doi.org/ 10.1016/j.enconman.2006.08.024. [17] Borghi M, Milani M, Paoluzzi R. Stationary axial flow force analysis on compensated spool valve. Int J Fluid Power 2000;1:17–25. [18] Yang YS, Semini C, Tsagarakis NG, Caldwell DG, Zhu YQ. Water hydraulics – a novel design of spool-type valves for enhanced dynamic performance. In: Proceeding of the 2008 IEEE/ASME international conference on advanced intelligent mechatronics, Xi’an; July 2008. p. 1308–14. [19] Miller R, Fujii Y, McCallum J, Strumolo G, Tobler W, Pritts C. CFD simulation of steady-state flow forces on spool-type hydraulic valves. SAE Technical Paper Series; 1999. p. 295–307. [20] Krishnaswamy K, Li PY. On using unstable electrohydraulic valves for control. ASME J Dyn Syst Meas Control 2002;124:183–90. [21] Du XW, Zou J, Fu X, Ji H, Yang HY. Effect of throttling grooves structure on cavitation noise. J Zhejiang Univ (Eng Sci) 2007;41(3):456–65. [22] Fang WM, Cheng LL, Fu X, Zhou H, Yang HY. Investigation on steady-state flow force of spool valve with U-grooves. J Zhejiang Univ (Eng Sci) 2010;44(3):576–80. [23] Borghi M, Milani M, Paoluzzi R. Influence of notches shape and number of notches on the metering characteristics of hydraulic spool valves. Int J Fluid Power 2005;3:5–18. [24] Chen JS, Liu XH, Yuan WR, Wang TJ, Zhao F. Dynamic characteristics of typical hydraulic notches. J Southwest Jiaotong Univ 2012;47(2):325–32. [25] Zhou S, Xu B, Yang HY. Flow force compensation of high speed on/off valve. Chin J Mech Eng 2006;42:5–8.