PARAMETRIC MODELLING OF AN AUTOMOTIVE HYDRAULIC VALVE N. Mittwollen, P. Tirgari Robert Bosch GmbH, D-70049 Stuttgart Tel.: +49 711 811 6946, Email: [email protected] Abstract. The modelling of a spring controlled hydraulic poppet valve with a chamfered seat is proposed. Non-linear effects of the transition from laminar attached to turbulent separated flow on flow rate and flow forces are taken into account in a consistent way by simple algebraic expressions. These expressions are based on Reynolds principle of similarity, and they shall possess general validity. They contain three non-dimensional parameters, which can easily be determined for a broad range of operating conditions by a limited amount of measurements for a given valve design. The model is applied to a hydraulic system simulation, where very stiff and discontinuous phenomenons significantly determine the dynamic system characteristic. The simulation results are verified by a comparison with measured pressure transients.

Introduction Due to the interactions between mechanical and hydraulic forces and the complex nature of fluid flows, the modelling and behavioural simulation of automotive hydraulic systems still remains a formidable challenge, especially in the higher frequency region (f > 1 kHz). Strong non-linearities in the governing conservation equations of mass and momentum, cavitation effects, wave propagation, laminar, transitional and turbulent flow, discontinuities, and stiff system equations have to be tackled within reasonable integration time limits. A popular approach is the use of tabulated flow data from stationary measurements or three dimensional CFD simulations of single hydraulic components. For a given system, accurate results can be achieved with this approach. However, if the hydraulic system characteristic is to be improved over a wide range of operating conditions by an optimization of valve parameters, such as spring preload and stiffness or maximum valve lift, more generally valid parametric models should be used. Integrating a three dimensional flow model into the behavioural simulation is possible in principle and would provide a complete set of parameters, but is impracticable due to unreasonable demands on computation times. Thus, an appropriate model has to be derived that contains only those parameters that are needed in the optimization process, plus those describing the operating conditions or the special valve design (e.g., valves with conical or spherical poppets or disc valves). To find such models, much insight into the typical flow effects is required, as can be found in [5,7] and as is summarized in [3]. The principles of similarity laws can be applied to get generally valid expressions. A well known example is the orifice flow rate characteristic (see Figure 1) [2,4]. It contains two generally valid non-dimensional parameters: the flow 2 ( pA – pB ) a α Q = αA h ----------------------------coefficient α, which depends on the Reynolds ρ number Re of the flow through the orifice. This model can be used for determining the flow rate of a αt hydraulic valve. The valve characteristic is significantly influenced by flow forces. They are usually taken into account only by an estimation of the momentum force of the inlet flow and of the turbulent separated Re cr jet flow, assumed to be exiting the valve seat [1,3,5,8]. This is insufficient for valves with chamRe fered seats. For operating conditions at low Reynolds numbers below or around 500, the flow at the narrow Figure 1: Orifice flow rate [4]. seat gap can be laminar and completely attached or transitional as well. A very simple approach to A similar paper was presented at the 2nd MATHMOD in Vienna [12]

account for the corresponding transitional flow effects is the use of an effective stiffness [6]. Since its validity is rather limited, the authors propose to describe the modelling approach to build a more general transitional flow force model for poppet valves, which is analogous to the above mentioned flow rate model.

Hydraulic System Pressure and flow rate nodes for pipe model P4=P C

Pressure source Inlet valve

P3 Q3

Q4

Outlet valve

P2

P1 Q2

Q1

pB Back flow line Pump element Damping chamber

Pressure control valve

Restrictor 2

Restrictor 1

Figure 2: Hydraulic system model (Bathfp) The investigated hydraulic system, displayed in Figure 2, is used as a low cost compact pressure source with the supply pressure p B . The hydraulic fluid is delivered into the system by a single piston reciprocating pump with inlet and outlet check valves. A high volumetric efficiency is guaranteed by a preloading pressure source. The pulsating pressure transients are coarsely smoothed by the compressibility of the fluid in the hydraulic damping chamber in combination with the restrictor 1 orifice. The level of the operating system pressure p B is determined by the pressure control valve. An extended back flow line delivers the redundant fluid to a reservoir, which is open to the atmosphere.

Parametric Model for the Pressure Control Valve The pressure control valve is a spring controlled poppet valve with a ball as the closing body and a chamfered seat. The proposed simulation model for this valve (Figure 3 left) takes into account the dependence of the cross-sectional orifice area (simplified expression)

π γ A h = --- h  D + h sin --- sin γ 2  2

(1)

on the valve lift h (γ = seat chamfer angle, D = ball diameter) for computing the flow rate Q with the orifice equation

Q = α ( Re )A h 2∆p ⁄ ρ ,

(2)

where ∆p = p inlet - p outlet is the pressure difference over the valve, and ρ is the fluid density. The flow rate coefficient α(Re) is allowed to change linearly with the square root of the Reynolds number Re = Q d h/A h ν of the flow through the orifice for Re < Re cr, as shown in Figure 1. Here, ν is the kinematic fluid viscosity, and d h = 2 h sin(γ/2) is the hydraulic diameter of the cross-sectional orifice area. The parameters to be fitted to flow rate measurements or to CFD simulations are the flow rate coefficient α t for turbulent separated flow conditions and the critical Reynolds number Re cr, identifying the transition from laminar to turbulent flow. A smoothing function is used in the transition region. All relevant static and dynamic forces, acting on the ball, are to be considered as well. Thus, the equation of motion

mx˙˙ + d D x˙ + cx = F h – F 0

(3)

with the valve ball mass m and lift x = h, the damping coefficient d D , the spring stiffness c and preload F 0 is used for the movement of the valve ball (Figure 3 left). The hydraulic force F h can be determined by carrying out a hydraulic force balance on a control volume around the valve ball, considering pressure and momentum forces (Figure 3 right):

+

ρQ ( v inlet – v outlet cos ( γ ⁄ 2 ) )

            

A p ( p inlet – p outlet )

        

Fh =

(4)

momentum force F m

pressure force F p

d D x˙

F0 + c x

Control volume

mx˙˙ x, x˙, x˙˙

Separated jet flow

D As

γ/2

Ah

x=h

voutlet p outlet

Ab

γ Fh

Q

Fh

Q

v inlet p inlet

Figure 3: Control valve model (left) and hydraulic force control volume (right) The flow enters the control volume with an average velocity of

v inlet = Q ⁄ A b

(5)

and leaves it with the velocity

v outlet =

2∆p ⁄ ρ ,

(6)

estimated by Bernoulli‘s equation [1,3,5]. The area A p that the pressure difference ∆p is acting on is usually set to a constant value (e.g., to the cross-sectional area A b = (π/4)D b 2 of the inlet bore for sharpedged seats [3,8], or to A s = (π/4)D 2 cos 2 (γ/2) for chamfered seats, Figure 3 left). This model is insufficient, because it assumes constant pressures p inlet over the area A p and p outlet over the remaining areas. However, unknown pressure distributions, strongly varying with flow conditions, develop in the valve seat region and significantly influence the force balance and thus the correct value of the hydraulic force F h . The authors propose to take this effect into account by making the area A p (now called effective pressurized area), and thus the pressure force F p as well, depend on these flow conditions, represented by the seat flow Reynolds number Re, in a similar way as the flow coefficient α does:

 A l + Re ⁄ Re cr ( A t – A l ) for Re ≤ Re cr (laminar flow) Ap =  . for Re > Re cr (turbulent flow)  At

(7)

For zero flow rate (Re = 0), i.e., for a closed valve, the above mentioned assumption of constant pressures over the two regions, separated by the seat‘s theoretical sealing line, is correct. In that case, the effective

pressurized area A p is equal to

A l = max ( A s , A b ) ,

(8)

whereas for a turbulent flow Re > Re cr, A p is set to

At = f p ⋅ Al .

(9)

Hydraulic force

Flow rate Q

One additional non-dimensional parameter to be fitted to experimental data is the turbulent effective pressurized area factor f p (0 < f p < 1), describing the influence of the relative pressure distribution in the valve seat gap. Fitted values for f p can be considerably below 1 due to relaModel tive sub-pressures, developing in the seat gap [5,7]. This parameter is mainly influMeasurement enced by the valve geometry. Equation (7) provides for a continuous transition of the effective pressurized area A p between the 0. corresponding values for the two states of 0. zero flow (only the hydrostatic pressure Valve lift h force is effective), i.e., A l and turbulent separated flow, i.e., A t, respectively. This transition is found to follow the same square root dependency on the seat flow Reynolds number Re as the flow coeffiPressure force F p cient α(Re) follows (Figure 1). Α comparison of model calculations with Measurement measurements, where operational ball lifts h were always smaller than the seat Model F h =F p +F m chamfer length, demonstrates this relationship (Figure 4). The reason for the change of the effective pressurized area A p at low Reynolds numbers Re up to the critical value Re cr is 0. that the relative pressure distribution in the seat gap is determined by the strongly Momentum force F m changing ratio of momentum to viscous forces in the flow, represented by the Rey0. Valve lift h nolds number [11]: At very low Reynolds numbers, viscous forces dominate in the flow, and the pressure drops from p inlet to Figure 4: Comparison of measured and calculated valve flow rate and hydraulic forces (∆p = constant). p outlet near the narrowest seat gap and is almost constant to either p inlet or p outlet around it. With an increasing Reynolds number, the rising influence of momentum forces in the flow make this pressure drop move towards the seat inlet, in addition to making substantial relative sub-pressures develop in the seat gap [5,7]. Furthermore, flow separation is caused. A comparison of the two hydraulic force components F p and F m in Figure 4 shows that the change in the hydraulic force F h is mainly determined by the change in the pressure force F p for Re < Re cr. At high Reynolds numbers Re > Re cr, momentum forces dominate over viscous forces in the flow. Thus, it is reasonable to set the effective pressurized area A p to a constant value there, which is smaller than for a laminar flow, and let the change in the momentum force F m determine the change in the hydraulic force F h alone. 0

. 0

Further Component Models The pump element is modelled as a hydraulic volume with a sinuous variation of its size. For the other hydraulic components of the circuit (Figure 2), standard models of the commercial simulation package Bathfp [9, 10], used in this investigation, are employed. For the back flow line, the Bathfp finite volume distributed model with each 4 internal pressure and flow rate nodes is used to allow for pressure wave dynamics. Cavitation and air release are taken into account by a two-phase homogeneous equilibrium model.

Simulation Results and Model Verification The operation of the hydraulic system (Figure 2) results in pressure transients, displayed in Figure 5. The comparison of the transients, determined experimentally (left) and by simulation (right), shows that all relevant hydraulic effects are accounted for by the simulation model. The pronounced pulsation of the system pressure p B is caused by the volume pulses, produced by the reciprocating pump. The pressure control valve operates at Reynolds numbers mostly below Re cr = 500, and the described effects of laminar-turbulent flow transition, with the further parameters α t = 0.7 and f p = 0.8, on hydraulic force and flow rate mainly determine the large amplitude and transient shape of the system pressure p B . These effects act on the closing body like an additional, strong stiffness. The level of the system pressure p B is mainly determined by the spring preload F 0 and slightly influenced by the transitional flow effects. Superimposed on the system pressure pulsation are bursts of pressure oscillations with a high frequency. These bursts are correlated with spikes in the pressure transient p c, downstream of the control valve, which are more pronounced at every second pump cycle. A further analysis of the system transients reveals that the origin of these peculiar pressure transients is the periodic creation and breakdown of cavitation in the extended back flow line, causing fluid column separations and reformations [6]. Simulation (Bathfp)

Experiment Upstream of pressure control valve pB

160

150

50 bar

40 140

50 bar

10 Pressure [bar]

Upstream of pressure control valve pB

50

120 30

100 5

20

100 80

Pipe node P4 = pC

60

50

10

pC

0

Downstream of pressure control valve pC pB

0

40 20

pC pB 0.12

0.14

0.16

0.18

0.2

Time [s]

Time Figure 5: Pressure transients

Conclusions The proposed simulation model is an attempt to map the strong influence of transitional flow phenomena, observed in valves with chamfered seats, on simple algebraic expressions, which are consistent for the flow rate and the hydraulic force, utilizing the similarity principle of fluid mechanics. A set of three adjusted parameters, the turbulent flow coefficient α t, the critical Reynolds number Re cr, and the newly introduced turbulent effective pressurized area factor f p , suffices to describe the static characteristic of the investigated poppet valve design with a spherical closing body over a wide range of operating conditions. Experimentally verified simulation results with this model also demonstrate its dynamic suitability.

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