An Experimental Study on the Use of Unstable ... - Me.umn.edu
An Experimental Study on the Use of Unstable Electrohydraulic Valves for Control 1 Qinghui Yuan and Perry Y. Li 2 Department of Mechanical Engineering University of Minnesota 111 Church St. SE Minneapolis MN 55455 Ps
Abstract
Pt
L1
L2
Sleeve
Single stage valves have their main spools stroked directly by solenoid actuators. They are cheaper and more reliable. Their use, however, is restricted to low bandwidth and low flow rate applications due to the limitation of the solenoid actuators. In a previous paper, a way proposed to alleviate the need for large solenoids in single stage valves by inducing spool instability using the transient flow forces, hence, improve the spool agility. In this paper, we study the underlying premise that the transient flow forces can be controlled by the “damping length”. Models for both steady and transient flow forces that include viscous effects are developed. Various models are analyzed, compared using CFD analysis and correlated to experiments. It was found that the model for the spool dynamics using the various flow force models are consistent with experimental results as long as viscous effects are taken into account. Both magnitude studies and experiments show that “damping lengths” and transient flow forces significantly affect spool agility. CFD studies also indicate that viscosity is an important factor to consider while modeling fluid flow forces. 1 Introduction Electrohydraulic valves are used to control flow in a hydraulic system. For high bandwidth, high flow rate applications, the force needed to actuate the spool that meters the flow into and out of the valve becomes significant. Because of the force/power limitation of solenoid actuators that are used to stroke the spool in single stage valves, single stage valves perform poorly in high bandwidth, high flow rate applications. In these situations, multi-stage valves are generally used. In these, the spools are driven by one or more pilot stage hydraulic valves. However, multistage valves tend to be more expensive to manufacture, more sensitive to dirt accumulation, and therefore are less reliable than single stage valves. Our research aims to improve the bandwidth and flow capabilities of single stage valves by alleviating the need 1 Research
supported by NSF-ENG/CMS. author
2 Corresponding
Xv Fe
Spool
Q
Q
Actuator
Figure 1: A four way direction flow control valve. for large and expensive solenoid. Our approach is to utilize unstable transient flow forces to increase the agility of the spool, so that less force/power is needed from the solenoid. The system will be stabilized by closed loop control. In a previous study [2], we demonstrated, using analysis and computer simulation, that the valves configured to be unstable can have faster step responses under solenoid saturation than stable counterparts. Also less positive power but more negative (braking) power is required to track a sinusoidal flow rate. A basic premise of the study [2] is that the sign of the “damping length”, which is a geometric dimension, determines whether the transient flow force is stabilizing or destabilizing. Instability due to transient flow forces has not been explicitly studied since 1960s [3, 1]. Moreover, the goal of studying transient flow forces in [3] and others have generally been to eliminate rather than to utilize the instability. The objectives of the paper are to 1) compare the basic flow force model in the literature against a model developed using computational fluid dynamics (CFD), with the inclusion of viscosity; 2) validate, experimentally, the hypothesis that damping lengths do affect the agility of the spool, under limited stroking capability of the solenoid actuator. The rest of paper is organized as follows. In section 2, we derive several models for the steady state and transient flow forces for incompressible, viscous fluid. Section 3 presents the experimental apparatus that is used to verify the effect of transient forces on spool dynamics. In section 4, steady state and transient flow p. 1
h
g
Xv
g
c θ
e
F` sleeve
d
Pl
x
Pr a
i
nal fluid velocity. (2) can be decomposed into a component associated with steady flow, and a component associated with transient flow: Z d 0 (3) ρv · dV . Fspool = −Flux + Fsleeve + − | {z } dt V | {z } steady flow force
f
b
F`spool a
b
Q
transient flow force
L1
Previous models for steady flow force [1, 3, 2] do not consider the viscosity effect, i.e. Frod = Fsleeve = 0. If fluid is assumed to enter the chamber in a vena contracta at an angle θ and a homogeneous speed Vin , and to leave the chamber perpendicularly to the spool, then, ρ Flux = ρQVin cos θ = Q2 cos θ (4) Cd Ao (xv )
Figure 2: Meter-in valve chamber. e θ
g
h
g
d
f
c
Xv
F`sleeve Pl
x
Pr a
i Q
b
F` spool a
b
L2
Figure 3: Meter-out valve chamber. forces are computed and estimated using CFD, and compared with the traditional analytical approaches. Section 5 presents the dynamic model for the spool incorporating the flow forces and solenoid dynamics. Section 6 presents some experimental results. Section 7 contains some concluding remarks. 2 Modeling of Fluid Flow Forces A typical four way directional control valve (Figure 1) contains a meter-in chamber (left) and a meter-out chamber (right). Let us consider associated with the meter-in and the meter-out chambers separately. Consider first the chamber with an orifice that is metering flow into the valve (Fig. 2). The force Fspool (positive to the right) that the spool experiences from the fluid can be calculated in various ways. Most fundamentally, it is given by: Fspool = Fland + Frod = A(Pr − Pl ) + Frod
(1)
where Fland is the pressure force acting on the lands, Frod is the viscous force acting on the spool rod, Pr and Pl are the pressures on the right and left lands, and A is the annulus area of the lands. Fspool can also be computed to be the reaction of the forces that the spool acts on the fluid. To wit, consider the longitudinal momentum equation for the fluid volume a-b-c-d-e-f-g inside the valve chamber. Z d 0 0 0 ρv·dV +Flux = Fspool +Fsleeve = −Fspool +Fsleeve dt V (2) 0 Here, Fspool = −Fspool is the reaction force that the 0 spool lands act on the fluid, Fsleeve = −Fsleeve is the viscous force that valve sleeve acts on the fluid, and Flux is the longitudinal momentum flux (towards the right) associated with the fluid entering and leaving the chamber, ρ is the fluid density, v is the longitudi-
where Q is the flow rate. The vena contracta area for a spool displacement xv can be written as Cd Ao (xv ) in which Cd is the discharge coefficient and Ao (xv ) is the orifice area. Also, by flow continuity, the total volume flow rate across any cross-sectional area perpendicular to the spool to the right of the inlet, is Q, therefore, the chamber fluid momentum is given by: Z M= ρv · dV = ρL1 Q, V
where L1 is the length between the outlet and the orifice. Hence, the steady and and unsteady flow forces are respectively given by: ρ 0 Fsteady = − Q2 cos θ + Fsleeve (5) Cd Ao (xv ) d (6) Ftransient = ρL1 Q. dt To analyze the transient flow forces, the quasi-static assumption is now utilized . This assumption states that at any inlet and outlet pressures Pin , Pout , the flow rate Q is determined by the spool displacement xv , Q(xv ). One such relationship is the orifice equation: s 2∆P Q(xv ) = Cd Ao (xv ) . (7) ρ where ∆P = Pin − Pout . 0 In the steady state, the viscous force Fsleeve is also a 0 function of the flow rate Q, i.e. Fsleeve (Q). For example, for Hagen-Poiseuille laminar flow [3], 0 (Q) = αµL1 Q Fsleeve
(8)
where α > 0 is some geometry dependent coefficient and µ is the dynamic viscosity. If the orifice equation (7) and laminar flow equation (8) are assumed, we have s # " 2∆P Fsteady = −Cd Ao (xv ) 2∆P cos θ − αµL1 ρ (9) s 2∆P dAo ∂Q x˙ v = ρL1 Cd x˙ v . (10) Ftransient = ρL1 ∂xv ρ dxv p. 2
8.5 in. 4.6 in.
0.25 in.
0.5 in.
Since Ao (xv ) increases monotonically with xv , the first term (momentum flux) and the second term (sleeve force), in the steady state flow force Fsteady acts like a spring forces, with positive and negative spring rates respective, to close or open the orifice. In particular, when the momentum flux effect dominates, the steady flow force is stabilizing. The transient flow force, Ftransient , for the meter-in valve chamber in Fig. 2, on the other hand, generates a negative damping, and hence an unstable effect on the spool.
A
B
C
D
E
F
G
H
I
J
K
Figure 5: Configuration of valve. LBK Xv
B
K
x
If the analysis above is done on a meter-out valve chamber (Fig. 3), then the steady and transient flow forces will be given by: ρ 0 Q2 cos θ + Fsleeve Cd Ao (xv ) d = −ρL2 Q, dt
Fsteady = − Ftransient
L FK
F
K
x
(11) Figure 6: Adjustable damping length. (12)
where L2 is the distance between the entry port and 0 the meter-out orifice in Fig. 3. Fsleeve = −αµL2 Q if laminar flow is observed. Thus, the transient flow force induces a positive damping, and hence stabilizing effect on the spool. The momentum flux and sleeve force terms in the steady flow force will still act like springs with positive and negative spring constants.
agility is improved. The experimental apparatus in Figure 4 consists of a custom built valve that allows for different (both positive and negative) damping lengths L. The spool is actuated by a pair of solenoids (in this paper, only one is used). Its position is measured by a linear potentiometer. The control of the solenoid and the position recordings are coordinated by a PC.
Since the spool is acted on by both meter-in and meterout chambers (Fig. 1), the net force that acts on the spool is then the sum of the forces. In particular, if the so-called “damping length” L := L2 − L1 is positive, the damping coefficient is positive, and if L < 0, the damping coefficient is negative. In commerical valves, L is designed to be positive. Our research is to choose L < 0 so as to improve the agility and responsiveness of the spool.
The core of the setup is a spool valve with the adjustable damping length L. The sleeve is made of acrylic resin so that the flow can be inspected visually. Adjustable damping lengths are achieved by having eleven (11) ports with male quick couplers attached, evenly spaced along the axis of the sleeve Figure 5. By completing the hydraulic circuit at different ports, different damping lengths are achieved.
In this paper, we study primarily the flow force for a single meter-in / meter-out chamber. Thus, the L = −L1 when studying meter-in chamber and L = L2 when studying the meter-out chamber. 3 Experimental apparatus The objective of the experiment is to confirm, as suggested by Eq.(6), that the damping length L can indeed manipulate both the direction and the magnitude of Ftransient , and specifically that by setting L < 0 spool
Result recorded for analysis Computer
Measurement System
Control Signal
Power amplifier
Solenoid Actuator
Valve
Figure 4: Experimental Schematic.
The spool consists of a hardened precision anodized aluminum threaded shaft and several bronze lands. The lands can be fixed at positions along the shaft that correspond to the chosen ports in order to achieve either a meter-in or a meter-out chamber. For example, if in Fig. 5, supply pressure is connected to port “B” and the port “K” is connected to the reservoir and is metered by a land, a meter-out chamber with damping length LBK is constructed in Fig. 6. Alternatively, if port “K” is the metered port connected to the supply pressure, and port “B” is connected to reservoir, then a meter-in chamber with damping length −LBK is obtained. By using ports “F” and “K”, damping lengths ±LF K are obtained. The valve is also equiped with ports for connection to an actuator. However, these are not used in this paper. To minimize the asymmetric lateral forces that will contribute to friction [3], circumferential grooves are machined on each land to equalize the pressure. Hydraulic fluid used is Mobil DTE46 that is fed by a pressure compensated pump from reservoir to the valve, and then to reservoir. Additional pressure relief valves and needle valves are also used to control the pressure in hydraulic circuit. The circuit in figure 7 is used to drive the solenoid (Ledex 173921-023) which is connected to the drain of p. 3
inlet
Vd
Vo
Rm
outlet
Ls
Df
rod
Is
Rs D Vg
G S
Figure 8: Grid for fluid model. Figure 7: Power amplifier and measurement system. Ls , Rs denotes the inductance and the resistance of the coil
the N-type MOSFET. A flyback diode Df is required to handle the voltage spike. The MOSFET functions as a (gate) voltage controlled current source for the solenoid. By adjusting the input voltage to the MOSFET, solenoid force can be controlled to simulate force / power limitations. When a step voltage is applied to the gate of the MOSFET, the spool is actuated. The start time Ts is monitored by a on-off switch mounted on the end of the spool. The spool displacement trajectory, especially the travel time Te is measured by the potentiometer. In addition, the solenoid coil current Is is also recorded. This allows the solenoid force trajectory to be calculated. Experimental results will be given in section 6. 4 CFD analysis of flow forces 4.1 Numerical model of valve Steady state CFD is used to determine flow forces and flow patterns that can better capture the effects of valve geometry. They provide alternatives to simple orifice equations in calculating flow forces as in section 2. These in turn can be used to estimate transient flow forces using the quasi-static assumption. The 3D computational volume used includes the chamber and both the entry and exit ports. In this paper, clearance is not considered. This might contribute to inaccurate jet angle at the vena contracta for small spool displacement. The geometry and mesh generation, hexahedral elements in cooper scheme, are implemented in GAMBIT. 72454 nodes and 61877 elements are generated, as can be seen in figure 8. The incompressible Navier-stokes equations without body forces are given by [4]: Continuity: ∇·V =0 (13) Momentum: ρ
∂V + ρV · ∇V = −∇P + µ∇2 V ∂t
Table 1: Steady state flow force xv (in.) A0 (m2 ) Q(m3 /s) θ(◦ ) 0 Fsleeve (n) Frod (n) A(Pr − Pl )(n) Fsteady |1 (n) Fsteady |2 (n) Fsteady |3 (n) Fsteady |4 (n)
0.025 1.61e-6 3.8e-5 80 -0.01 0.01 -0.29 -0.28 -0.35 -0.56 -0.23
0.075 8.37e-6 2.07e-4 69 -1.64 1.08 -4.90 -3.83 -4.18 -4.25 -2.47
0.1 1.16e-5 2.94e-4 69 -2.40 1.60 -7.24 -5.64 -6.09 -5.90 -3.42
ary conditions consists in imposing the inlet pressure Ps = 689475.7P a (100psi) and outlet pressure Pt = 101300P a (1 atm). No slip conditions are imposed on all land faces, rod and sleeve walls. Fluid density of ρ = 871kg/m3 , and dynamic viscosity µ = 0.0375kg/ms(40◦ C) are used. Four geometries corresponding to spool displacements xv of 0.025,0.05,0.075 and 0.1 in. are considered. The positive damping length L = 0.216m is used. Eqs.(13)-(14) are solved using the first order Upwind scheme until steady condition is achieved. The solutions are obtained using FLUENT 5.5 on the IBM SP supercomputer at the University of Minnesota. 4.2 Estimation of steady flow forces CFD analysis provide the fluid velocity field and the pressure distribution. From these, the viscous force on the rod Frod , and pressure Pl , Pr acting on the lands can be calculated. These provide the various methods for estimating steady state flow forces in section 2 to be compared. Most fundamentally, Eq. (1) gives: Fsteady |1 = A(Pr − Pl ) + Frod .
(15)
Second, CFD can be used to obtain a numerical relationship which is alternative to the orifice equation for the function Q(xv ). Then Eq.(5) derived using momentum theory is:
(14)
where ρ is fluid density, V is the fluid velocity, P is the pressure and µ is dynamic viscosity. The SIMPLE pressure correction approach [4] is applied to decouple the continuity and momentum equations. Bound-
0.05 4.52e-6 1.18e-4 74 -0.92 0.60 -2.50 -1.90 -2.09 -1.98 -1.02
Fsteady |2 = −
ρ 0 Q2 cos θ + Fsleeve Cd Ao (xv )
(16)
0 where the sleeve force Fsleeve is obtained from CFD, Cd = 0.6 is the discharge coefficient, the orifice area
p. 4
Table 2: Transient flow force
xv (in.) Ao (m2 ) Q(m3 /s) dQ/dA0 (m/s) dA0 /dxv (m) Ftransient |1 (n) Ftransient |2 (n)
0 0 0 -
0.025 1.61e-6 3.8e-5 24.1615 0.0038 -4.1234 -4.0609
0.05 4.52e-6 1.18e-4 27.1821 0.0051 -6.2259 -5.4502
0.075 8.37e-6 2.07e-4 23.1169 0.0058 -6.0215 -6.1983
0.1 1.16e-5 2.94e-4 26.9350 0.0062 -7.4999 -6.6257
Ao (xv ) is given: D02
µ
¶
(D0 − 2xv )(D0 − xv )xv D0 − 2xv − D0 D0 (17) where D0 is the diameter of the orifice.
Ao (xv ) =
4
acos
The jet angle θ is obtained from figure 9. Since the numerical model does not consider the clearances between the lands and the sleeve, the θ deviates from the the traditional theory [3] for small xv is small. For sufficiently large xv , however, θ ≈ 69◦ as predicted in [3]. The third estimation was the laminar flow assumption and the orifice equation. Eq.(9) is given by: s " # 2∆P Fsteady |3 = −Cd Ao (xv ) 2∆P cos θ + αµL ρ (18) where α = 1.1e − 6/m2 , and Cd = 0.6. α is estimated from the CFD data, ∆P = Ps − Pt . If the sleeve force is not considered then the steady flow force will be Fsteady |4 = −Cd Ao (xv )2∆P cos θ. Therefore for L > 0, this underestimates the steady flow force, and for L < 0, it overestimated it. The comparison of the steady flow force computed using different methods results are shown in table 1. 4.3 Estimation of transient flow forces It is difficult to use CFD to directly estimate transient flow forces because CFD hardly achieves the dynamic transmogrification of the volume. Instead, the transient flow forces is estimated using the quasi-static assumption in section 2 to give Eq. (6): ∂Q ∂Ao Ftransient |1 = ρL x˙ v ∂Ao ∂xv
(19)
dQ o where ∂A ∂xv is obtained from (17), dAo is estimated from the numerical solution Q(xv ) at various xv ’s. Using the orifice to determine Q(xv ), Eq.(10) gives an alternate model of the transient flow force: s 2∆P dAo x˙ v (20) Ftransient |2 = ρLCd ρ dxv
Figure
9:
Estimation of jet angles 0.025, 0.05, 0.075, 0.1 in.
for
xv
=
that the Eqs.(18) and (20) which are derived using the simple orifice equations, should provide relatively accurate flow force estimation compared with the CFD solutions. The deviation for small xv is probably due to error in estimating the jet angle because clearance is not considered. In section 5, Eqs.(18) and (20) will be used to determine the spool dynamics. The comparison studies also show that Fsteady and Ftransient are of the same order of magnitude, which verifies the feasibility of improving the spool agility by using the transient flow force. It is also remarkable that the viscous forces Fsleeve , Frod , which are neglected in previous studies, are significant and have the same order of magnitude as Fsteady . In fact, if L = −0.216m, then the steady flow force is −1.01n. This shows that the viscous force compensates for a large proportion of the steady flow forces. 5 Spool dynamics The spool dynamics, taking into account the flow forces, and the solenoid are given by: Mx ¨v = Fsteady + Ftransient + Fe + Ff
(21)
where M is the total mass of the spool and the armature of the solenoid, Fsteady and Ftransient are the steady and transient flow forces given by Eqs. (18) and (20), and Ff is the friction and is chosen to be 2N . Fe is the solenoid force given by: Fe =
λ2φ N 2 i2 2µ0 Aa (Ka xv + Kb )2
(22)
where N is the number of turns in the coil, i is the solenoid current, Ka and Kb are reluctance related parameters that are identified from manufacturer’s performance data using the method in [5], λφ is flux leakage coefficient, µ0 is permeability of free space, Aa is cross-section area of air gap. Given the solenoid current trajectory, i(t), the spool displacement trajectory can be predicted by integrating Eq.(21). Eq.(21) will be verified by comparing its prediction with experimental result in section 6.
The results for x˙ v = 0.254m/s (obtained from experiments) are shown in table 2.
6 Experimental Results
Tables 1 and 2 demonstrate that except for small orifice openings (xv = 0.025in), results from different models are well within 10% of each other. This suggests
The main purpose of the paper is to verify that the transient flow force can be used to improve the agility of the spool, especially when the solenoid force is limited. p. 5
48
−3
10
x 10
Vg=2.3V Vg=2.4V
46
L= −0.118m 44
8
42
L=0.216m
6
xv(m)
response time(ms)
L= −0.216m
40 38 36
L=0.118m 4
34 32 30
2
28 −0.25
−0.2
−0.15
−0.1
−0.05 0 0.05 damping length(m)
0.1
0.15
0.2
0.25
0 Experimental Calculated −2 20
30
40
50 time(ms)
60
Figure 11: Response time vs. damping length 70
80
−3
8
x 10
L= −0.216m 7
Figure 10: Experimental and model predicted displace-
L= −0.118m
6
ment curves for Vg = 2.3V
5
L= 0.118m
To do this, we investigate the time the spool takes to travel a full range of 0.0094m from an initial solenoid air-gap of 0.015m, at damping lengths of 0.216, 0.118, -0.118, -0.216 m, and using input gate voltages Vg = 2.2V , 2.3V , 2.4V .
xv(m)
4
3
1
0
−1
The step responses to Vg = 2.3V for the various damping lengths are shown in Figure 10. Clearly, as the damping length decreases the transit time decreases. Superimposed on Fig. 10 is also the spool displacement prediction using the spool dynamics (21). Notice that the prediction is close to the experiment result, indicating that the flow force and solenoid force models capture the actual situations. The transit times for Vg = 2.3V and Vg = 2.4V for the different damping lengths are shown in Fig. 11. In both cases, the transit times decrease as damping length decreases. However, the damping length has a more significant effect for Vg = 2.3V than for Vg = 2.4V as indicated by the slopes in Fig.11. This result was predicted by the numerical analysis in [2] which shows that spool instability has a greater effect on spool agility when the solenoid limitation is more severe. In fact when Vg = 2.2V , the solenoid limitation is so severe that the spool is halted by friction and is unable to travel the entire stroke length (figure 12). However, a longer stroke is attained when the damping length is smaller. For L = −0.216m, the transient flow force is able to assist the spool to nearly complete the entire stroke. 7 Conclusions The paper investigates flow forces of a control valve for a incompressible but viscous hydraulic fluid. Steady flow force models based on orifice equation is quite comparable to those computed based on direction CFD measurement and momentum analysis. However, viscous effect has been shown to be quite significant. This is surprising to us, we plan on investigating this further. The dynamic model of the spool computed based on solenoid equation, and flow force models is quite ac-
L= 0.216m
2
0
20
40
60
time(ms)
80
100
120
Figure 12: Displacement curve for Vg = 2.2V curate in predicting the experimental results. Experimental investigation of the effect of damping lengths confirm that by decreasing the damping lengths, the agility of the spool, as measured by transit times, improves, especially when the limitation on the solenoid is more severe. These results highlight the importance of unstable transient flow force in high bandwidth / high flow rate applications. References [1] J. F. Blackburn, G. Reethof, and L. L. Shearer. Fluid Power Control. MIT Press, 1960. [2] K. Krishnaswamy and P. Y. Li. On using unstable electrolydraulic valves for control. In 2000 ACC Proceedings, May 2000. Also to appear in ASME Journal of Dynamic Systems, Measurement and Control. [3] H. E. Merritt. Hydraulic Control System. John Wiley and Sons, 1967. [4] J. C. Tannehill, D. A. Anderson, and R. H. Pletcher. Computational fluid mechanics and heat transfer. Taylar and Francis, Philadephia, PA, 1997. [5] Y.Xu and B.Jones. A simple means of predicting the dynamic response of electromagnetic actuators. Mechatranics, 7(7):589–598, 1997.
p. 6
Abstract
Pt
L1
L2
Sleeve
Single stage valves have their main spools stroked directly by solenoid actuators. They are cheaper and more reliable. Their use, however, is restricted to low bandwidth and low flow rate applications due to the limitation of the solenoid actuators. In a previous paper, a way proposed to alleviate the need for large solenoids in single stage valves by inducing spool instability using the transient flow forces, hence, improve the spool agility. In this paper, we study the underlying premise that the transient flow forces can be controlled by the “damping length”. Models for both steady and transient flow forces that include viscous effects are developed. Various models are analyzed, compared using CFD analysis and correlated to experiments. It was found that the model for the spool dynamics using the various flow force models are consistent with experimental results as long as viscous effects are taken into account. Both magnitude studies and experiments show that “damping lengths” and transient flow forces significantly affect spool agility. CFD studies also indicate that viscosity is an important factor to consider while modeling fluid flow forces. 1 Introduction Electrohydraulic valves are used to control flow in a hydraulic system. For high bandwidth, high flow rate applications, the force needed to actuate the spool that meters the flow into and out of the valve becomes significant. Because of the force/power limitation of solenoid actuators that are used to stroke the spool in single stage valves, single stage valves perform poorly in high bandwidth, high flow rate applications. In these situations, multi-stage valves are generally used. In these, the spools are driven by one or more pilot stage hydraulic valves. However, multistage valves tend to be more expensive to manufacture, more sensitive to dirt accumulation, and therefore are less reliable than single stage valves. Our research aims to improve the bandwidth and flow capabilities of single stage valves by alleviating the need 1 Research
supported by NSF-ENG/CMS. author
2 Corresponding
Xv Fe
Spool
Q
Q
Actuator
Figure 1: A four way direction flow control valve. for large and expensive solenoid. Our approach is to utilize unstable transient flow forces to increase the agility of the spool, so that less force/power is needed from the solenoid. The system will be stabilized by closed loop control. In a previous study [2], we demonstrated, using analysis and computer simulation, that the valves configured to be unstable can have faster step responses under solenoid saturation than stable counterparts. Also less positive power but more negative (braking) power is required to track a sinusoidal flow rate. A basic premise of the study [2] is that the sign of the “damping length”, which is a geometric dimension, determines whether the transient flow force is stabilizing or destabilizing. Instability due to transient flow forces has not been explicitly studied since 1960s [3, 1]. Moreover, the goal of studying transient flow forces in [3] and others have generally been to eliminate rather than to utilize the instability. The objectives of the paper are to 1) compare the basic flow force model in the literature against a model developed using computational fluid dynamics (CFD), with the inclusion of viscosity; 2) validate, experimentally, the hypothesis that damping lengths do affect the agility of the spool, under limited stroking capability of the solenoid actuator. The rest of paper is organized as follows. In section 2, we derive several models for the steady state and transient flow forces for incompressible, viscous fluid. Section 3 presents the experimental apparatus that is used to verify the effect of transient forces on spool dynamics. In section 4, steady state and transient flow p. 1
h
g
Xv
g
c θ
e
F` sleeve
d
Pl
x
Pr a
i
nal fluid velocity. (2) can be decomposed into a component associated with steady flow, and a component associated with transient flow: Z d 0 (3) ρv · dV . Fspool = −Flux + Fsleeve + − | {z } dt V | {z } steady flow force
f
b
F`spool a
b
Q
transient flow force
L1
Previous models for steady flow force [1, 3, 2] do not consider the viscosity effect, i.e. Frod = Fsleeve = 0. If fluid is assumed to enter the chamber in a vena contracta at an angle θ and a homogeneous speed Vin , and to leave the chamber perpendicularly to the spool, then, ρ Flux = ρQVin cos θ = Q2 cos θ (4) Cd Ao (xv )
Figure 2: Meter-in valve chamber. e θ
g
h
g
d
f
c
Xv
F`sleeve Pl
x
Pr a
i Q
b
F` spool a
b
L2
Figure 3: Meter-out valve chamber. forces are computed and estimated using CFD, and compared with the traditional analytical approaches. Section 5 presents the dynamic model for the spool incorporating the flow forces and solenoid dynamics. Section 6 presents some experimental results. Section 7 contains some concluding remarks. 2 Modeling of Fluid Flow Forces A typical four way directional control valve (Figure 1) contains a meter-in chamber (left) and a meter-out chamber (right). Let us consider associated with the meter-in and the meter-out chambers separately. Consider first the chamber with an orifice that is metering flow into the valve (Fig. 2). The force Fspool (positive to the right) that the spool experiences from the fluid can be calculated in various ways. Most fundamentally, it is given by: Fspool = Fland + Frod = A(Pr − Pl ) + Frod
(1)
where Fland is the pressure force acting on the lands, Frod is the viscous force acting on the spool rod, Pr and Pl are the pressures on the right and left lands, and A is the annulus area of the lands. Fspool can also be computed to be the reaction of the forces that the spool acts on the fluid. To wit, consider the longitudinal momentum equation for the fluid volume a-b-c-d-e-f-g inside the valve chamber. Z d 0 0 0 ρv·dV +Flux = Fspool +Fsleeve = −Fspool +Fsleeve dt V (2) 0 Here, Fspool = −Fspool is the reaction force that the 0 spool lands act on the fluid, Fsleeve = −Fsleeve is the viscous force that valve sleeve acts on the fluid, and Flux is the longitudinal momentum flux (towards the right) associated with the fluid entering and leaving the chamber, ρ is the fluid density, v is the longitudi-
where Q is the flow rate. The vena contracta area for a spool displacement xv can be written as Cd Ao (xv ) in which Cd is the discharge coefficient and Ao (xv ) is the orifice area. Also, by flow continuity, the total volume flow rate across any cross-sectional area perpendicular to the spool to the right of the inlet, is Q, therefore, the chamber fluid momentum is given by: Z M= ρv · dV = ρL1 Q, V
where L1 is the length between the outlet and the orifice. Hence, the steady and and unsteady flow forces are respectively given by: ρ 0 Fsteady = − Q2 cos θ + Fsleeve (5) Cd Ao (xv ) d (6) Ftransient = ρL1 Q. dt To analyze the transient flow forces, the quasi-static assumption is now utilized . This assumption states that at any inlet and outlet pressures Pin , Pout , the flow rate Q is determined by the spool displacement xv , Q(xv ). One such relationship is the orifice equation: s 2∆P Q(xv ) = Cd Ao (xv ) . (7) ρ where ∆P = Pin − Pout . 0 In the steady state, the viscous force Fsleeve is also a 0 function of the flow rate Q, i.e. Fsleeve (Q). For example, for Hagen-Poiseuille laminar flow [3], 0 (Q) = αµL1 Q Fsleeve
(8)
where α > 0 is some geometry dependent coefficient and µ is the dynamic viscosity. If the orifice equation (7) and laminar flow equation (8) are assumed, we have s # " 2∆P Fsteady = −Cd Ao (xv ) 2∆P cos θ − αµL1 ρ (9) s 2∆P dAo ∂Q x˙ v = ρL1 Cd x˙ v . (10) Ftransient = ρL1 ∂xv ρ dxv p. 2
8.5 in. 4.6 in.
0.25 in.
0.5 in.
Since Ao (xv ) increases monotonically with xv , the first term (momentum flux) and the second term (sleeve force), in the steady state flow force Fsteady acts like a spring forces, with positive and negative spring rates respective, to close or open the orifice. In particular, when the momentum flux effect dominates, the steady flow force is stabilizing. The transient flow force, Ftransient , for the meter-in valve chamber in Fig. 2, on the other hand, generates a negative damping, and hence an unstable effect on the spool.
A
B
C
D
E
F
G
H
I
J
K
Figure 5: Configuration of valve. LBK Xv
B
K
x
If the analysis above is done on a meter-out valve chamber (Fig. 3), then the steady and transient flow forces will be given by: ρ 0 Q2 cos θ + Fsleeve Cd Ao (xv ) d = −ρL2 Q, dt
Fsteady = − Ftransient
L FK
F
K
x
(11) Figure 6: Adjustable damping length. (12)
where L2 is the distance between the entry port and 0 the meter-out orifice in Fig. 3. Fsleeve = −αµL2 Q if laminar flow is observed. Thus, the transient flow force induces a positive damping, and hence stabilizing effect on the spool. The momentum flux and sleeve force terms in the steady flow force will still act like springs with positive and negative spring constants.
agility is improved. The experimental apparatus in Figure 4 consists of a custom built valve that allows for different (both positive and negative) damping lengths L. The spool is actuated by a pair of solenoids (in this paper, only one is used). Its position is measured by a linear potentiometer. The control of the solenoid and the position recordings are coordinated by a PC.
Since the spool is acted on by both meter-in and meterout chambers (Fig. 1), the net force that acts on the spool is then the sum of the forces. In particular, if the so-called “damping length” L := L2 − L1 is positive, the damping coefficient is positive, and if L < 0, the damping coefficient is negative. In commerical valves, L is designed to be positive. Our research is to choose L < 0 so as to improve the agility and responsiveness of the spool.
The core of the setup is a spool valve with the adjustable damping length L. The sleeve is made of acrylic resin so that the flow can be inspected visually. Adjustable damping lengths are achieved by having eleven (11) ports with male quick couplers attached, evenly spaced along the axis of the sleeve Figure 5. By completing the hydraulic circuit at different ports, different damping lengths are achieved.
In this paper, we study primarily the flow force for a single meter-in / meter-out chamber. Thus, the L = −L1 when studying meter-in chamber and L = L2 when studying the meter-out chamber. 3 Experimental apparatus The objective of the experiment is to confirm, as suggested by Eq.(6), that the damping length L can indeed manipulate both the direction and the magnitude of Ftransient , and specifically that by setting L < 0 spool
Result recorded for analysis Computer
Measurement System
Control Signal
Power amplifier
Solenoid Actuator
Valve
Figure 4: Experimental Schematic.
The spool consists of a hardened precision anodized aluminum threaded shaft and several bronze lands. The lands can be fixed at positions along the shaft that correspond to the chosen ports in order to achieve either a meter-in or a meter-out chamber. For example, if in Fig. 5, supply pressure is connected to port “B” and the port “K” is connected to the reservoir and is metered by a land, a meter-out chamber with damping length LBK is constructed in Fig. 6. Alternatively, if port “K” is the metered port connected to the supply pressure, and port “B” is connected to reservoir, then a meter-in chamber with damping length −LBK is obtained. By using ports “F” and “K”, damping lengths ±LF K are obtained. The valve is also equiped with ports for connection to an actuator. However, these are not used in this paper. To minimize the asymmetric lateral forces that will contribute to friction [3], circumferential grooves are machined on each land to equalize the pressure. Hydraulic fluid used is Mobil DTE46 that is fed by a pressure compensated pump from reservoir to the valve, and then to reservoir. Additional pressure relief valves and needle valves are also used to control the pressure in hydraulic circuit. The circuit in figure 7 is used to drive the solenoid (Ledex 173921-023) which is connected to the drain of p. 3
inlet
Vd
Vo
Rm
outlet
Ls
Df
rod
Is
Rs D Vg
G S
Figure 8: Grid for fluid model. Figure 7: Power amplifier and measurement system. Ls , Rs denotes the inductance and the resistance of the coil
the N-type MOSFET. A flyback diode Df is required to handle the voltage spike. The MOSFET functions as a (gate) voltage controlled current source for the solenoid. By adjusting the input voltage to the MOSFET, solenoid force can be controlled to simulate force / power limitations. When a step voltage is applied to the gate of the MOSFET, the spool is actuated. The start time Ts is monitored by a on-off switch mounted on the end of the spool. The spool displacement trajectory, especially the travel time Te is measured by the potentiometer. In addition, the solenoid coil current Is is also recorded. This allows the solenoid force trajectory to be calculated. Experimental results will be given in section 6. 4 CFD analysis of flow forces 4.1 Numerical model of valve Steady state CFD is used to determine flow forces and flow patterns that can better capture the effects of valve geometry. They provide alternatives to simple orifice equations in calculating flow forces as in section 2. These in turn can be used to estimate transient flow forces using the quasi-static assumption. The 3D computational volume used includes the chamber and both the entry and exit ports. In this paper, clearance is not considered. This might contribute to inaccurate jet angle at the vena contracta for small spool displacement. The geometry and mesh generation, hexahedral elements in cooper scheme, are implemented in GAMBIT. 72454 nodes and 61877 elements are generated, as can be seen in figure 8. The incompressible Navier-stokes equations without body forces are given by [4]: Continuity: ∇·V =0 (13) Momentum: ρ
∂V + ρV · ∇V = −∇P + µ∇2 V ∂t
Table 1: Steady state flow force xv (in.) A0 (m2 ) Q(m3 /s) θ(◦ ) 0 Fsleeve (n) Frod (n) A(Pr − Pl )(n) Fsteady |1 (n) Fsteady |2 (n) Fsteady |3 (n) Fsteady |4 (n)
0.025 1.61e-6 3.8e-5 80 -0.01 0.01 -0.29 -0.28 -0.35 -0.56 -0.23
0.075 8.37e-6 2.07e-4 69 -1.64 1.08 -4.90 -3.83 -4.18 -4.25 -2.47
0.1 1.16e-5 2.94e-4 69 -2.40 1.60 -7.24 -5.64 -6.09 -5.90 -3.42
ary conditions consists in imposing the inlet pressure Ps = 689475.7P a (100psi) and outlet pressure Pt = 101300P a (1 atm). No slip conditions are imposed on all land faces, rod and sleeve walls. Fluid density of ρ = 871kg/m3 , and dynamic viscosity µ = 0.0375kg/ms(40◦ C) are used. Four geometries corresponding to spool displacements xv of 0.025,0.05,0.075 and 0.1 in. are considered. The positive damping length L = 0.216m is used. Eqs.(13)-(14) are solved using the first order Upwind scheme until steady condition is achieved. The solutions are obtained using FLUENT 5.5 on the IBM SP supercomputer at the University of Minnesota. 4.2 Estimation of steady flow forces CFD analysis provide the fluid velocity field and the pressure distribution. From these, the viscous force on the rod Frod , and pressure Pl , Pr acting on the lands can be calculated. These provide the various methods for estimating steady state flow forces in section 2 to be compared. Most fundamentally, Eq. (1) gives: Fsteady |1 = A(Pr − Pl ) + Frod .
(15)
Second, CFD can be used to obtain a numerical relationship which is alternative to the orifice equation for the function Q(xv ). Then Eq.(5) derived using momentum theory is:
(14)
where ρ is fluid density, V is the fluid velocity, P is the pressure and µ is dynamic viscosity. The SIMPLE pressure correction approach [4] is applied to decouple the continuity and momentum equations. Bound-
0.05 4.52e-6 1.18e-4 74 -0.92 0.60 -2.50 -1.90 -2.09 -1.98 -1.02
Fsteady |2 = −
ρ 0 Q2 cos θ + Fsleeve Cd Ao (xv )
(16)
0 where the sleeve force Fsleeve is obtained from CFD, Cd = 0.6 is the discharge coefficient, the orifice area
p. 4
Table 2: Transient flow force
xv (in.) Ao (m2 ) Q(m3 /s) dQ/dA0 (m/s) dA0 /dxv (m) Ftransient |1 (n) Ftransient |2 (n)
0 0 0 -
0.025 1.61e-6 3.8e-5 24.1615 0.0038 -4.1234 -4.0609
0.05 4.52e-6 1.18e-4 27.1821 0.0051 -6.2259 -5.4502
0.075 8.37e-6 2.07e-4 23.1169 0.0058 -6.0215 -6.1983
0.1 1.16e-5 2.94e-4 26.9350 0.0062 -7.4999 -6.6257
Ao (xv ) is given: D02
µ
¶
(D0 − 2xv )(D0 − xv )xv D0 − 2xv − D0 D0 (17) where D0 is the diameter of the orifice.
Ao (xv ) =
4
acos
The jet angle θ is obtained from figure 9. Since the numerical model does not consider the clearances between the lands and the sleeve, the θ deviates from the the traditional theory [3] for small xv is small. For sufficiently large xv , however, θ ≈ 69◦ as predicted in [3]. The third estimation was the laminar flow assumption and the orifice equation. Eq.(9) is given by: s " # 2∆P Fsteady |3 = −Cd Ao (xv ) 2∆P cos θ + αµL ρ (18) where α = 1.1e − 6/m2 , and Cd = 0.6. α is estimated from the CFD data, ∆P = Ps − Pt . If the sleeve force is not considered then the steady flow force will be Fsteady |4 = −Cd Ao (xv )2∆P cos θ. Therefore for L > 0, this underestimates the steady flow force, and for L < 0, it overestimated it. The comparison of the steady flow force computed using different methods results are shown in table 1. 4.3 Estimation of transient flow forces It is difficult to use CFD to directly estimate transient flow forces because CFD hardly achieves the dynamic transmogrification of the volume. Instead, the transient flow forces is estimated using the quasi-static assumption in section 2 to give Eq. (6): ∂Q ∂Ao Ftransient |1 = ρL x˙ v ∂Ao ∂xv
(19)
dQ o where ∂A ∂xv is obtained from (17), dAo is estimated from the numerical solution Q(xv ) at various xv ’s. Using the orifice to determine Q(xv ), Eq.(10) gives an alternate model of the transient flow force: s 2∆P dAo x˙ v (20) Ftransient |2 = ρLCd ρ dxv
Figure
9:
Estimation of jet angles 0.025, 0.05, 0.075, 0.1 in.
for
xv
=
that the Eqs.(18) and (20) which are derived using the simple orifice equations, should provide relatively accurate flow force estimation compared with the CFD solutions. The deviation for small xv is probably due to error in estimating the jet angle because clearance is not considered. In section 5, Eqs.(18) and (20) will be used to determine the spool dynamics. The comparison studies also show that Fsteady and Ftransient are of the same order of magnitude, which verifies the feasibility of improving the spool agility by using the transient flow force. It is also remarkable that the viscous forces Fsleeve , Frod , which are neglected in previous studies, are significant and have the same order of magnitude as Fsteady . In fact, if L = −0.216m, then the steady flow force is −1.01n. This shows that the viscous force compensates for a large proportion of the steady flow forces. 5 Spool dynamics The spool dynamics, taking into account the flow forces, and the solenoid are given by: Mx ¨v = Fsteady + Ftransient + Fe + Ff
(21)
where M is the total mass of the spool and the armature of the solenoid, Fsteady and Ftransient are the steady and transient flow forces given by Eqs. (18) and (20), and Ff is the friction and is chosen to be 2N . Fe is the solenoid force given by: Fe =
λ2φ N 2 i2 2µ0 Aa (Ka xv + Kb )2
(22)
where N is the number of turns in the coil, i is the solenoid current, Ka and Kb are reluctance related parameters that are identified from manufacturer’s performance data using the method in [5], λφ is flux leakage coefficient, µ0 is permeability of free space, Aa is cross-section area of air gap. Given the solenoid current trajectory, i(t), the spool displacement trajectory can be predicted by integrating Eq.(21). Eq.(21) will be verified by comparing its prediction with experimental result in section 6.
The results for x˙ v = 0.254m/s (obtained from experiments) are shown in table 2.
6 Experimental Results
Tables 1 and 2 demonstrate that except for small orifice openings (xv = 0.025in), results from different models are well within 10% of each other. This suggests
The main purpose of the paper is to verify that the transient flow force can be used to improve the agility of the spool, especially when the solenoid force is limited. p. 5
48
−3
10
x 10
Vg=2.3V Vg=2.4V
46
L= −0.118m 44
8
42
L=0.216m
6
xv(m)
response time(ms)
L= −0.216m
40 38 36
L=0.118m 4
34 32 30
2
28 −0.25
−0.2
−0.15
−0.1
−0.05 0 0.05 damping length(m)
0.1
0.15
0.2
0.25
0 Experimental Calculated −2 20
30
40
50 time(ms)
60
Figure 11: Response time vs. damping length 70
80
−3
8
x 10
L= −0.216m 7
Figure 10: Experimental and model predicted displace-
L= −0.118m
6
ment curves for Vg = 2.3V
5
L= 0.118m
To do this, we investigate the time the spool takes to travel a full range of 0.0094m from an initial solenoid air-gap of 0.015m, at damping lengths of 0.216, 0.118, -0.118, -0.216 m, and using input gate voltages Vg = 2.2V , 2.3V , 2.4V .
xv(m)
4
3
1
0
−1
The step responses to Vg = 2.3V for the various damping lengths are shown in Figure 10. Clearly, as the damping length decreases the transit time decreases. Superimposed on Fig. 10 is also the spool displacement prediction using the spool dynamics (21). Notice that the prediction is close to the experiment result, indicating that the flow force and solenoid force models capture the actual situations. The transit times for Vg = 2.3V and Vg = 2.4V for the different damping lengths are shown in Fig. 11. In both cases, the transit times decrease as damping length decreases. However, the damping length has a more significant effect for Vg = 2.3V than for Vg = 2.4V as indicated by the slopes in Fig.11. This result was predicted by the numerical analysis in [2] which shows that spool instability has a greater effect on spool agility when the solenoid limitation is more severe. In fact when Vg = 2.2V , the solenoid limitation is so severe that the spool is halted by friction and is unable to travel the entire stroke length (figure 12). However, a longer stroke is attained when the damping length is smaller. For L = −0.216m, the transient flow force is able to assist the spool to nearly complete the entire stroke. 7 Conclusions The paper investigates flow forces of a control valve for a incompressible but viscous hydraulic fluid. Steady flow force models based on orifice equation is quite comparable to those computed based on direction CFD measurement and momentum analysis. However, viscous effect has been shown to be quite significant. This is surprising to us, we plan on investigating this further. The dynamic model of the spool computed based on solenoid equation, and flow force models is quite ac-
L= 0.216m
2
0
20
40
60
time(ms)
80
100
120
Figure 12: Displacement curve for Vg = 2.2V curate in predicting the experimental results. Experimental investigation of the effect of damping lengths confirm that by decreasing the damping lengths, the agility of the spool, as measured by transit times, improves, especially when the limitation on the solenoid is more severe. These results highlight the importance of unstable transient flow force in high bandwidth / high flow rate applications. References [1] J. F. Blackburn, G. Reethof, and L. L. Shearer. Fluid Power Control. MIT Press, 1960. [2] K. Krishnaswamy and P. Y. Li. On using unstable electrolydraulic valves for control. In 2000 ACC Proceedings, May 2000. Also to appear in ASME Journal of Dynamic Systems, Measurement and Control. [3] H. E. Merritt. Hydraulic Control System. John Wiley and Sons, 1967. [4] J. C. Tannehill, D. A. Anderson, and R. H. Pletcher. Computational fluid mechanics and heat transfer. Taylar and Francis, Philadephia, PA, 1997. [5] Y.Xu and B.Jones. A simple means of predicting the dynamic response of electromagnetic actuators. Mechatranics, 7(7):589–598, 1997.
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