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Modelling of a Variable Displacement Lubricating Pump with Air Dissolution Dynamics.

Introduction

Variable displacement vane pumps with adjustable pressure setting represent the most advanced type of flow generation units for engine lubrication. The advantages in a driving cycle of modulating the pump displacement to satisfy the engine requirements are by now unquestionable [1].

The simplest type of displacement control consists in exploiting the oil pressure to generate on the pump stator a hydraulic force that is counterbalanced by a spring. The displacement reduction can be obtained through either a linear [2] or a rotational motion of the stator [3]. Regardless of the kinematic mechanism, a drawback of this solution is the effect of the pressure distribution on the stator track, which leads in some operating conditions to an excessive and undesired reduction of the pump displacement [4]. An effective solution is the use of a piloted control through a small hydraulic valve [5]. With this method, the pressure setting of the displacement control is practically insensitive to the internal forces [6]. A further improvement leading to a considerable fuel saving is constituted by the adoption of an active control of the circuit pressure. Beside a continuous modulation, as the solution presented by Burke et al. [7], a simpler but still effective method is the two-level approach, as the system analyzed in the reference [8]. It involves the use of a two-position electro-valve controlled by the engine ECU, with the aim of enabling or disabling the pilot stage in order to change the pressure setting of the displacement control. The correct design of such systems requires as input the knowledge of the force acting on the stator due to the pressure in all variable volume chambers.

The first comprehensive lumped parameter model of a variable displacement vane pump for engine lubrication with linear motion of the stator was presented in the early 2000s [9]. Other authors developed models with a pivoting stator ring. Cantore et al. [10] built a model of a pump for high-speed engines using the built-in libraries of the simulation environment LMS Amesim. Barbarelli et al. [11] developed a model in MATLAB Simulink[R] validated experimentally in terms of pressure history in a chamber. The circumferential pressure distribution on the stator track was also measured by Bianchini et al. [12]. A detailed mathematical model of a pump with pivoting stator ring was presented by Geist and Resh [13] as well as by Truong et al. [14]. A 1D approach was instead used by Harrison et al. [15] with the software GT-SUITE[R].

The very first CFD model of a vane pump was developed by Jiang and Perng [16], later the specialized software PumpLinx[R] was developed [17] and used by various researchers. Wang et al. [18] studied a pump with a pivoting stator and a direct acting displacement control. An in-depth analysis on the internal forces generated by the pressure distribution on the stator track was carried out by Sullivan and Sehmby [19]. Later Frosina et al. [20] simulated a vane pump provided with a piloted displacement control. The same research group, together with other scholars, developed also a 3D model of a radially balanced vane pump for power split transmissions [21].

All these studies have focused on the modelling of the pump itself, but a little attention has been paid to the influence of the air release/dissolution processes on the pump performance.

In the early fifties Schweitzer and Szebehely [22] demonstrated that the air dynamics can be described by a first order function. Over the years, this concept has been implemented in fluid models where the transport and the dynamics of the gaseous phase are taken into account. For instance, time constants for air release and re-solution were used by Jiang et al. [23] for the simulation of the pressure wave propagation in a pipeline and by Borghi et al. [24] for studying the meshing zone of an external gear pump. Zhou et al. [25] developed a customized fluid model applied to the same machine type, in which the time derivative of the separated air is also function of the pressure relative to the saturation value. Such a model has been recently updated with a better mass conservation of the different phases and applied to the simulation of a gerotor pump [26].

In CFD models the non-homogeneous distribution of the air/vapour content in the variable chambers can be determined. For instance, del Campo et al. [27] studied the air fraction evolution in an external gear pump using a 2D model developed in ANSYS Fluent[R]. The software PumpLinx implements the Full Cavitation Model developed by Singhal et al. [28]. However 2D and 3D models are not suitable for system-level simulations due to the unacceptable computational times.

In this paper, a detailed lumped parameter model of a lubricating vane pump with air dynamics is presented. The model, implemented in the LMS Amesim environment, has been validated in conditions of high and low aeration levels. Aim of the study is to demonstrate how it is possible to obtain a reliable fast running model able to work in very critical operating conditions characterized by extremely high free air content. Moreover, the influence of the gas fraction on both steady-state and dynamic performances has been analyzed. Finally, the influence of the time constants for the air release/dissolution processes has been evaluated.

Component Description

The pump used as reference is a 19 cc/rev vane type unit provided with a two-level pressure control of the displacement. A view of the pump with the indication of the ports is shown in Figure 1, while in Figure 2 the hydraulic scheme of the pump control is reported.

The displacement is varied through the linear movement of the stator ring, which can slide with respect to the housing, in order to modify the eccentricity with the shaft. Its prismatic guides work also as linear actuators with different active surfaces. The delivery pressure acts always on the actuator with smaller surface "a" generating a force that tends to decrease the pump displacement. On the opposite side, the hydraulic force generated by a larger actuator with area "A" and the force exerted by a spring tend to maintain the maximum displacement.

The pressure [p.sub.x] acting on the surface A is modulated by a small pressure relief valve RV with a remote pilot. On the engine, the pick-up point of the pilot line is located directly in the main gallery of the lubricating circuit, namely downstream from the oil filter.

When the electro-valve EV is closed (configuration OFF), as long as the pressure in the circuit remains lower than the pressure setting of the relief valve [p.sub.2]*, the delivery pressure acts on both actuators, but thanks to the larger surface A, the pump is maintained at the maximum displacement. While if the circuit pressure reaches the value [p.sub.2]*, a small control flow generates a pressure drop in the calibrated orifice R, allowing the reduction of the modulated pressure [p.sub.x] and the movement of the stator. Once the stator and the valve spool are in hydraulic equilibrium, the pressure in the circuit is maintained ideally equal to [p.sub.2]*. This configuration is suitable in severe operating conditions, for instance with high oil temperature and high load, in order to guarantee a significant flow rate for the journal bearings and for the piston cooling jets even when the circuit resistance is low.

On the contrary the electro-valve EV, when energized (configuration ON), allows venting the chamber with surface A. In this case [p.sub.x] ideally coincides with the atmospheric pressure and the only force that is able to maintain the pump at maximum displacement is generated by the spring with equivalent pressure setting [p.sub.1]* < [p.sub.2]*. Hence, when the valve is activated, the reduction of the pump displacement occurs when the circuit pressure achieves the lower setting [p.sub.1]*. This configuration allows a power saving in non-critical operating conditions.

Pump Model

Pressure and Flow Rate Evaluation

In a lumped parameter model, the pump is divided in control volumes to which the mass conservation in isothermal conditions is applied:

[dp/dt]=[B/V]([summation][Q.sub.i]-[omega][dV/d[phi]]) Eq. (1)

being p the pressure, V the volume, [beta] the fluid bulk modulus, [Q.sub.i] the flow rate through the generic port i-th (positive if ingoing), [omega] and [phi] respectively the angular speed and the position of the shaft. In this specific case, the control volumes are associated to the seven variable chambers and to a number of fixed capacities representing the volume of the pipes. Different methods can be used for the evaluation of the angular derivative of the volumes; however, for the vane pumps the vector rays approach [9] is particularly suitable, since it allows obtaining the exact analytic expression. The volume of the chambers is obtained by numerical integration of the derivative. The control volumes are connected by hydraulic resistances, representing the flow areas S in the port plate and the leakage passageways. For the former the Equation 2 for flow in turbulent regime condition is used:

Q=[C.sub.d]S[square root of [2[DELTA]P/[rho]]] Eq. (2)

where [DELTA]p is the pressure drop, [rho] the fluid density and [C.sub.d] the discharge coefficient.

All geometrical quantities, such as volumes, their derivatives and flow areas are function of both the shaft angle and the current eccentricity. With reference to Figure 3, the main flow area is constituted by the frontal surface given by the overlap between the chamber contour and the port plate profile. Such an area is calculated automatically using a procedure that reads the coordinates of the port profile from a file built directly from the CAD drawing by means of an AutoLISP routine [29]. The area is calculated for different positions of the shaft with a step of 0.5 degrees and for five positions of the stator, from zero to maximum eccentricity. The data are stored in a 2D matrix that is interpolated linearly during the simulation.

Additional contributions to the flow area, not function of the eccentricity, are given by the lateral passages obtained by reducing the axial height of the stator on both sides and, for the delivery side only, by the groove on the pump casing. These flow areas have been calculated manually and added to the frontal ones. In Figure 4 the geometric quantities necessary for the implementation of the Equations 1 and 2 are shown for the maximum eccentricity.

Leakages Evaluation

All possible leakage paths in the rotary assembly are shown in Figure 5. The quantities defining the geometry of the gaps are calculated analytically as function of the main pump parameters.

Each variable chamber is connected to the volume beneath the vanes through the axial clearance between the rotor and the cover (leakage [Q.sub.11]) and through the clearance between the vanes and the rotor slots (leakage [Q.sub.12]). In this last case, it is assumed that the vanes are in contact against the rotor on the trailing side, as represented in Figure 5. The flow rate is calculated with the Equation 3 valid for laminar flow through a constant-height rectangular gap:

Q=[[b.sub.c][h.sup.3.sub.c]/12[mu][l.sub.c]][DELTA]P Eq. (3)

being [b.sub.c], [h.sub.c] and [l.sub.c] respectively the width, height and length of the gap and [mu] the fluid dynamic viscosity.

Each variable chamber communicates with the adjacent ones through the axial gap between the cover and the vane (leakage [Q.sub.13]). In this case, the width of the gap is variable, being coincident with the vane lift h. The volume underneath the vanes is connected to the sump through the clearance of the shaft bearing (leakage [Q.sub.14]). The flow rate is calculated with the Equation 4 valid for laminar flow in an annulus between a shaft and a cylinder [30]:

[Q.sub.14] =[[pi][R.sub.sh][c.sup.3]/6[mu][l.sub.c]](1 + 1.5[[epsilon].sup.2])[P.sub.r] Eq. (4)

where c is the radial clearance and [epsilon] the eccentricity ratio of the bearing.

The pressure [p.sub.r] in the volume underneath the vanes is calculated with the Equation 1 and it is function of the leakages [Q.sub.l1], [Q.sub.l2] and [Q.sub.l4]. Its value, intermediate between the delivery and the suction pressure, influences the radial equilibrium of the vanes with a consequence on the leakage on the tip [Q.sub.l5]. In Figure 6, the main geometric quantities of a vane in a generic angular position [phi] are shown; [O.sub.1] and [O.sub.2] are respectively the center of the rotor and of the stator. At a generic real vane lift h, the current minimum distance between the vane tip and the stator track is:

[h.sub.min]=([h.sub.th]-h)cos[delta] Eq. (5)

where [h.sub.th] is the theoretical vane lift that is calculated from the kinematics assuming the contact between the vane tip and the stator track [9] and the angle [delta] is:

[delta]=arcsin([esin [phi]/[R.sub.s]-[r.sub.v]]) Eq.(6)

The current eccentricity e is calculated by the equilibrium of the stator. The gap height [h.sub.x] in correspondence of the generic point K belonging to the vane tip and identified by the angle [[delta].sub.x] is:

[mathematical expression not reproducible] Eq. (7)

In a reference system integral with the vane, the stator slides with a velocity [v.sub.0]. The abscissa x, with origin in correspondence of the minimum height [h.sub.min], identifies a generic cross section of the gap. It is defined as:

[mathematical expression not reproducible] Eq. (8)

The flow rate per unit width crossing the orifice in a generic section with height [h.sub.x] is calculated with the more general equation for laminar flow through variable height gaps with sliding surfaces:

[Q.sub.l]=[[v.sub.0][h.sub.x]/2]-[[h.sup.3.sub.x]/12[mu]]*[dp/dx] Eq. (9)

The flow rate does not depend on the coordinate x, therefore the pressure gradient can be written as function of a constant [h.sub.0] as follows:

[dp/dx]=[6[mu][v.sub.0]/[h.sup.2.sub.x]](1-[[h.sub.0]/[h.sub.x]]) Eq. (10)

The equation must be integrated between -[x.sub.min] where the pressure is [p.sub.j] and [x.sub.max] where the pressure is [p.sub.j-1], in order to determine the constant [h.sub.0]:

[h.sub.0] = [[I.sub.1]/[I.sub.2]-[[DELTA]P/6[I.sub.2][mu][v.sub.0]] Eq. (11)

being [I.sub.1] and [I.sub.2] function of the gap profile. Such constants are determined by numerical integration and depend on the current lift h:

[mathematical expression not reproducible] Eq. (12)

Finally, the Equation 9 can be rewritten as:

Q = [[h.sub.0][v.sub.0]/2]=[[v.sub.0]/2]*[[I.sub.1]/[I.sub.2]] Eq. (13)

Since for high values of the gap [h.sub.min] the flow regime can be turbulent and in this case the Equation 13 overestimates the flow rate, the leakage [Q.sub.l5] is calculated with the Equation 14:

[mathematical expression not reproducible] Eq. (14)

where Ci is the discharge coefficient that would be obtained with the Equation 2 with the flow rate calculated with the Equation 13:

C =-Q- i (r) Eq. (15)

Finally, the leakages between the inlet and delivery volumes and the spring chamber where the modulated pressure px acts are also considered using equivalent rectangular gaps with height equal to the axial clearance between the stator and the casing.

Stator Equilibrium

For the evaluation of the internal forces on the stator, three contributions are considered (Figure 7):

* the force Fp exerted directly by the pressure in each variable chamber,

* the reaction forces of the vanes FRa against the stator track,

* the centrifugal force of the oil in the chambers.

The pressure force [F.sub.p] is calculated starting from the length of the segment 12 connecting the theoretical contact points between two consecutive vanes and the stator. The force is applied in the midpoint of the segment with direction identified by the angle [xi] calculated as function of the shaft angle. The resultant along the x-axis of all forces contributes to the equilibrium of the stator together with the forces of the actuators and of the spring.

For the calculation of the reaction force [F.sub.Ra] the equation of equilibrium is applied to determine the radial position of the vane. In case of detachment from the stator track, the tip clearance and the corresponding leakage flow is evaluated. With reference to Figure 8, the axial and tangential force on the vane due to the pressure are respectively:

[mathematical expression not reproducible] Eq. (16)

where H is the axial height of the rotary assembly. While, called [m.sub.v] the mass of the vane, the centrifugal force is expressed as:

[F.sub.c] = [m.sub.v][[omega].sup.2] ([R.sub.r]+h-[[l.sub.v]/2] Eq. (17)

The forces acting on the vane perpendicularly to its axis are:

* the force [F.sub.pt] due to the pressure difference between the two chambers,

* the reaction forces [F.sub.R1] and [F.sub.R2],

* the friction force on the vane tip [F.sub.ft]

The reaction force at the lower radius can be evaluated from the equilibrium of the moments with respect to the point where the force [F.sub.R2] is applied (the contribution of the axial forces is neglected):

[F.sub.R1]=[h/[l.sub.v]-h]*([[F.sub.pt]/2]+[F.sub.ft])Eq. (18)

The friction forces are evaluated in a simplified way as: [mathematical expression not reproducible] Eq. (19)

where [f.sub.p] and [f.sub.v] are respectively the pressure and velocity dependent friction coefficients.

The reaction force at the outer radius can be obtained from the equilibrium of the vane in the direction perpendicular to the axis:

[F.sub.R2] = [F.sub.R1] + [F.sub.pt] + [F.sub.ft] Eq. (20)

The contact force [F.sub.Ra] is:

[mathematical expression not reproducible] Eq. (21)

The calculation of the reaction force [F.sub.Ra] does not take into account the thrust exerted between the roots of the vanes through the floating rings. This simplification can be considered reasonable by considering that the force [F.sub.Ra] is normally one order of magnitude smaller than the pressure force [F.sub.P] For the evaluation of the friction on the vane tip, [f.sub.p] = 0.1 was used; it is coherent with the values available in literature [31, 32].

It must be noted that, since the pressure [p.sub.r] is lower than the delivery pressure, the centrifugal force is not able to keep the vane in contact with the stator track when the speed is low and the vane is located on the delivery side. Nevertheless such a condition is not critical if the vane separates two chambers both connected to the delivery volume. However, depending on the specific geometry of the port plate and on the operating conditions (speed, oil temperature, displacement), if a pressure peak occurred in the chamber during the transition from the suction to the delivery side, the vane would detach with a detrimental effect on the volumetric efficiency, but it is not the case.

Fluid Model

A hydraulic fluid always includes some gas, mostly air, which is either dissolved or separated (undissolved). Depending on the system's evolution, air changes from one form to the other one. Indeed, when the pressure increases in the hydraulic circuit, dissolution phenomenon occurs and gas bubbles (previously existing in equilibrium with the saturated liquid) tends to collapse and to dissolve in the liquid. Conversely if the pressure decreases, aeration phenomenon occurs, thus gas bubbles tends to be released from the liquid and to grow. It is well established from literature that only the separated air modifies the fluid properties, as depicted in Figure 9 with the LMS Amesim fluid model.

The Henry-Dalton law gives the volumetric amount of air that is dissolved in the liquid (at reference temperature) as linear function of the pressure level, the slope of the curve being the well-known Bunsen coefficient. Moreover, it assumes that aeration and dissolution are instantaneous phenomena; this implicitly means that the amount of dissolved air is always in equilibrium within the liquid. However, in fast acting systems, such as hydraulic pumps, the high pressure derivative does not let enough time to the air for reaching equilibrium conditions; consequently the amount of separated air is not only barotropic but also time-dependent.

The thermal-hydraulic model of fluid in LMS Amesim offers different options to take into account aeration phenomenon, depending on the simulation purposes. Assumptions and conservation equations corresponding to each aeration model are described more in details in the reference [33].

With the most advanced model, suitable for full dynamics analysis, the total and undissolved air amounts are independent variables computed from corresponding mass fractions conservation laws applied to each volume, including hydraulic lines. The evolution of undissolved gas mass fraction [x.sub.u] takes into account dynamics for aeration or dissolution considering a first order lag characterized by a user-defined time constant [tau], as from Siemens PLM [34]:

[[dx.sub.u]/dt] = [1/V*[rho]]([summation]d[mx.sub.u]-[x.sub.u]*[summation]dm)+[[x.sub.u,eq]-[x.sub.u]/[tau]] Eq. (22)

Time constants [tau] can be differently set for dissolution and aeration:

* if the current undissolved fraction is greater than the equilibrium value given by Henry's law ([x.sub.u] > [x.sub.u,eq])qi, the gas progressively dissolves within the liquid to reach the equilibrium state;

* on the contrary, if the current undissolved fraction is lower than the equilibrium value ([x.sub.u] < [x.sub.u,eq]), the gas is released in the liquid in form of bubbles.

As alternative, different user-defined functions can be also implemented for the aeration and dissolution rate, however it was found that the second term of the Equation 22 gives satisfactory results, as shown later.

For most gases and liquids, aeration is a much faster phenomenon than dissolution and consequently the time constant for dissolution is much greater than for aeration. The saturation line (Figure 10) gives the pressure level required to completely dissolve the total air content into the liquid; its slope [alpha] corresponds to the Bunsen coefficient (for mineral oil [alpha] = 0.09).

The explicit relationship (23) between the saturation pressure [p.sub.sat] and the current amount of total gas mass fraction [x.sub.g] can be obtained considering the following:

* the Henry-Dalton law, defining the partial pressure of a gaseous phase above the solution as function of the molar fraction,

* the Bunsen coefficient, defining the percent of volume of the gas dissolved in a unit volume of the fluid at normal conditions,

* and finally, the assumptions of ideal gas and negligible number of moles of dissolved gas with respect to the liquid:

[P.sub.sat]=[[[rho].sub.1]R[T.sub.0]/[alpha]]*[[x.sub.g]/[M.sub.air]] Eq. (23)

As example, an ideal compression and decompression step is applied to a closed volume, such as a trapped chamber within a hydrostatic pump, with the aim of inducing dissolution and aeration events. The total air mass fraction within the volume is constant, 230 mg/kg, with a corresponding saturation pressure [p.sub.sat] = 1.77 bar absolute to completely dissolve the air into the liquid.

Figure 11 depicts simulation results corresponding to a compression/decompression step crossing the saturation pressure and obtained considering two different time constants: 1 ms for aeration and 5 s for dissolution. Simulation highlights the occurrence of a transient pressure peak corresponding to the air dissolution into the liquid, from the initial equilibrium conditions (1 barA, 80 [degrees]C) to new conditions following the compression step. The amount of undissolved air at equilibrium depends on the instantaneous pressure level with respect to the saturation pressure (Figure 10) and vaporization pressure. Due to the dissolution in progress along the time, the volume occupied by the liquid is slightly increasing and consequently the pressure smoothly reduces depending on the set time constant.

In a similar way, aeration is induced by an expansion step occurring at 11 sec; consequently, the air previously dissolved in the liquid in equilibrium (1.3 barA, 80 [degrees]C) starts releasing till reaching new equilibrium conditions (corresponding to the initial state). Due to the small time constant set for aeration, the transient conditions are much faster than in the case of dissolution.

Test Rig

The experimental measurements were performed on a test bench specific for lubricating pumps at Pierburg Pump Technology - Livorno plant. The pump under test PU1 was driven by a variable speed motor M1 through a sprocket-chain system. The hydraulic test circuit was intentionally simplified with respect to the standard layout in order to have a more accurate comparison with the simulation of the pressure ripple. The delivery line was constituted by a straight rigid pipe, whose qualitative cross section is shown in Figure 12. The load was simulated by a calibrated circular orifice with a diameter of 6.1 mm representing the resistance of the lubricating circuit. An additional restrictor, with diameter 8.5 mm, was used to simulate the pressure drop generated by the fluid conditioning system, namely the filter and the heat exchanger of the engine. The pilot port of the displacement control was connected to the volume between the two restrictors by means of a rigid copper pipe with internal diameter of 5 mm and length 0.3 m. Two piezo-resistive pressure transducers KISTLER with measuring range 0-50 bar and accuracy 0.1 bar sensed the delivery and the pilot pressure.

A photo of the pump mounted on the test rig is shown in Figure 13. The pump sucked the oil from a 40 L reservoir. The fluid conditioning system was constituted by electric elements, immersed directly in the reservoir as shown in Figure 13, and a chiller for cooling the oil through an auxiliary pump PU3. Both heater and cooler were controlled in closed-loop in order to maintain the desired temperature with an accuracy of [+ or -]2[degrees]C. The feedback signal was the output of a temperature detector in the reservoir. The main properties of the working fluid with SAE grade 5W20 are listed in Table 1.

A gerotor pump PU2, with the suction port partially connected to the atmosphere, was used as source of air flow rate. The delivery port was connected through a hose on the bottom of the reservoir at a distance of about 40 cm from the suction pipe of the pump under test.

The amount of separated air in the reservoir was measured by means of an optically based method through the commercial instrument Air-X from the company Delta Services Industriels [35]. The principle is the density measurement based on the absorption capacity of the X-rays by the oil [36]. The fluid sample is continuously taken from the reservoir by means of an internal pump and sent to the measuring chamber where the X-rays are emitted. On the opposite side of the chamber, a detector measures the amount of radiation that is increased by a higher fraction of air bubbles. A probe is used to compensate the variation of the pressure and of the temperature. The measuring range is from 0% to 100% of gas content. A self-calibration of the instrument was performed on the specific oil used for the test. Based on the acquisition time used for the tests, the accuracy in terms of volume gas content is of the order of 0.5%. The desired amount of separated air fraction was controlled manually through the speed of the motor M2.

Complete Simulation Model

The complete model of the pump and of the test rig is reported in Figure 14. The main components of the rotating assembly of the pump have been developed at the Politecnico di Torino with the Ameset tool in C code and they comply with the Thermal Hydraulic Library standard of LMS Amesim. The customized components are the following:

* 1: implements the Equation 1 for all variable chambers,

* 2: calculate inlet and outlet flow rates with the Equation 2,

* 3: calculates the inter-chambers leakages [Q.sub.l2], [Q.sub.l3] and [Q.sub.l5],

* 4: calculates the axial leakages [Q.sub.l1] and [Q.sub.l4],

* 5: represent a multiple hydraulic junction,

* 6: simulates the stator dynamics,

* 7: calculates the volumes and derivatives of the chambers,

* 8: calculates the frontal flow areas,

* 9: interpolate the lateral surfaces and the groove flow area

* 10: supplies the geometric parameters.

In the components 3 and 4 the gas transport is not implemented, i.e. the air crosses the variable chambers only through the inlet/outlet flow areas. The remaining elements of the pump are simulated by models belonging to the standard libraries. The leakages between the stator and the casing are simulated with the laminar gaps 11.

The delivery volume is simulated by a model of a hydraulic pipe 12 in which the resistive, capacitive and inertia effects are lumped in two elements with the same total volume of the real capacity. An identical model is used for the pipe 13 with length 130 mm (see Figure 12) and for the pipe 14 with length 90 mm. The former is divided in 4 subvolumes, the latter in two pipes with one element each. Hence, the entire test circuit is discretized with eight elements with a mean length of about 35 mm. It was found that the use of a higher number of subdivisions does not give significant improvements. The discharge coefficients of the restrictors have been left to the default value of 0.7. The pilot line 15 is simulated by a pipe with five internal nodes and frequency dependent friction.

The simulations were performed in isothermal conditions, hence the equations of energy balance for evaluation of the fluid temperature have been deactivated. The typical computational time is about 10 s per shaft revolution on a single core of an eight-core Xeon HT processor at 3.4 GHz.

Results and Discussion

Model Validation

The importance of the leakage models for the correct determination of the pump flow rate has been discussed in a previous study [37] performed on a similar unit. Nevertheless, it must be observed that an accurate evaluation of the leakages is crucial also in this paper, since the real flow rate influences the mean value of the pressure induced by the fixed restrictors shown in the scheme of Figure 12. Moreover, the mean pressure is influenced also by the current position of the stator that depends on the pressure history in the chambers and on the resultant of the forces exerted by the vanes. In turn, the former depends also on the flow areas, the latter is influenced by the detachment of some vanes. Although this paper is focused mainly on the influence of the oil aeration on the pump behavior, it is evident how all aspects of the pump itself must be taken into account in a very detailed way.

The experimental tests were executed at 80 [degrees]C, three angular speeds, two different configurations of the electro-valve, namely ON and OFF, and two different aeration levels, for a total of12 operating conditions. In particular, the configuration with high percentage of gas, indicated simply as "with air", was obtained by insufflating air into the reservoir in order to obtain a mixture with about 7% of free air in volume. The configuration with low aeration, simply identified as "no air", was obtained with the gerotor pump turned off and in this case the reading of the Air-X instrument was 0.1%. As far as the dissolved air fraction is concerned, it is assumed that the oil was always saturated with 9% of air, leading to a total gas fraction of 16% in the case "with air".

The list of the 12 configurations is reported in Table 2. In each configuration, once reached the steady-state conditions in terms of temperature and reading of the Air-X instrument, the pressure signals were acquired.

In the simulation model, to achieve the steady-state conditions of the undissolved gas mass fraction in the entire delivery line, at least 20-25 revolutions of the shaft are required. In fact it is necessary to wait that the air is transported from the reservoir up to the end of the delivery pipe. However, thanks to the hyper-threading feature of the processor (it is equivalent to have 16 logical cores) and to the batch run facility of the software, all configurations in Table 2 can be simulated simultaneously, each run on a different core, in a total time of about 5 min.

A sensitivity analysis was carried out for finding the values of the time constants for aeration and dissolution that allowed a good agreement with the experimental data. Best results can be obtained with [10.sup.-3] s for the aeration process and a time greater than 1 s for the dissolution for all configurations; in particular the presented results have been obtained with 5 s. The use of a high time constant for the dissolution process in hydraulic machines is in agreement with the results of Kim and Murrenhoff [38]. It is worth to notice that even better results could be obtained using different time constants for each configuration. However, it was preferred to find the values of better compromise for all configurations.

In Figure 15, the comparison between the simulated delivery pressure as function of the speed and the experimental points measured by the transducer P1 is reported. The simulated continuous lines have been obtained as a sequence of steady-state points with a speed increment of 100 rpm.

At 1474 rpm the pump works always at maximum displacement, hence the mean pressure value is imposed by the load, while at higher speed is decided by the displacement control.

With the piloted control (valve OFF) the displacement is reduced starting from 2400 rpm and such a value is not influenced by the presence of separated air. A maximum variation of about 0.4 bar is observed in the simulated regulated pressure between the tests with and without air. The same difference is also visible in the experimental data. On the contrary, with the direct acting control (valve ON) the maximum difference of the regulated pressure is greater than 1 bar and the same trend is observed experimentally. The tests bring to evidence the higher sensitivity to the operating conditions of the direct acting control with respect to the piloted one, as described more in details in the next section.

The delivery pressure ripple for six different configurations is shown from Figures 16 to 21. For sake of completeness, it must be highlighted that the relative positioning along the x-axis of the simulated and experimental curves has been made in order to have the coincidence of the main peaks, since a measurement of the absolute position of the shaft was not possible.

In order to evaluate the influence of the air content uncertainty, the simulations were also performed in different conditions. For instance, if it is assumed that the amount of dissolved air is 6%, i.e. the oil is not saturated, on average the mean value of the simulated delivery pressure in the tests "without air" increases of about 0.1 bar, with the worst case in the configuration 12 where the increment is 0.2 bar. The mean increment of the pressure amplitude is of the order of 0.15 bar. The configurations "with air" are less sensitive, since both amplitude and mean value are incremented by about 0.05 bar.

Another simulation considered a higher fraction of separated gas in the test "without air" to take into account the measurement error of the air-X instrument. If a percentage of 0.5% of free air is considered, instead of 0.1%, on average the mean simulated pressure decreases by 0.1 bar and the amplitude by 0.2 bar.

It is worth to mention that generally the amount of air in the oil has also a slight influence on the suction capability of the pump [39]. In this case however, the phenomenon of the incomplete chambers filling never occurred, since the tests at high speed were performed at low displacement.

Control Stability Issues

The pump, its control and the circuit constitute a feedback system that can potentially feature instabilities. A robust design needs to take into account also the dynamic behavior, in order to ensure that the system could always work in the stable region. The developed model is suitable for bringing to evidence instabilities and can be used as design tool for optimizing the dynamic response. It is known that the stability margin of a remotely piloted valve is lower than in a layout with a direct pilot. In a previous study [40] it was highlighted that the use of an external pilot line decreases drastically the cut-off frequency of the valve. Consequently, with a relatively low-frequency oscillating pressure signal, when the pressure at the inlet of the pilot pipe increases the valve closes instead of opening; vice versa when the pressure is decreasing. In the specific case of the displacement control, the consequence is an increment of the flow rate as the pressure rises, leading to an unstable behavior. The instability is also worsened by high values of the hydraulic capacitance V/[beta] of the delivery line, i.e. long pipes and high percentage of separated air.

The pump used for the current study was designed in order to work in the stable region when coupled with the lubricating circuit. However it is straightforward that an excessive increment of the delivery volume will lead inevitably to instability, above all when the effective fluid bulk modulus is very low.

In order to make this issue emerge, a transient simulation was performed with a delivery pipe between the two fixed restrictors 1.5 m long, instead of 90 mm. Starting from a speed of 1474 rpm, corresponding to maximum displacement, the velocity is incremented linearly up to 4474 rpm in 0.5 s. In Figure 22 the stator eccentricity and the speed are shown as function of the time. As expected, in the operating condition with high amount of separated air the instability of the stator is observed, with a high amplitude oscillation, about 1/3 of the entire stroke, and with low frequency, about 7 Hz. On the contrary, the pump remains stable in the configuration without initial separated air, unless the pipe is incremented up the unrealistic value of about 6 m.

Since the instability is generated by the oscillation of the pilot valve, no oscillation of the stator was observed in the configuration ON of the electro-valve, even with high amount of air and very long pipes; in fact in this case the pilot valve remains always closed.

Influence of the Time Constants

The time constants for dissolution and aeration have a strong influence on the instantaneous amount of separated air, which in turn determines the value of the effective bulk modulus of the air/oil mixture. Based on the Equation 1, it is evident how the consequence of a low bulk modulus is the reduction of the pressure time derivative with an influence on the pressure oscillations. However, a not negligible consequence also on the steady-state behavior is observed in Figure 15, above all in the configuration ON. In this case in fact, at high speed a variation of more than 1 bar occurs between the tests with and without air. The reason can be found from the analysis of the pressure distribution on the stator track.

In the theoretical case of ideal timing and ideal fluid, the pressure in the variable chambers rises instantaneously up to the delivery value as the chamber reaches the maximum volume and the connection with the outlet port opens. Similarly, the pressure decreases up to the suction value when the chamber is at minimum volume and it connects to the inlet port. With reference to Figure 23a, in this ideal situation the resultant force of the pressure distribution is directed vertically. In the real case, for optimization reasons, it is common practice to delay the connection with the outlet port.

Depending on the fluid conditions in the chamber that has just left the inlet side, two different cases must be considered. If all air is dissolved in the oil when the compression phase begins, a pressure peak occurs, leading to a component of the mean resultant force that tends to help the larger actuator to maintain the maximum displacement (Figure 23b). On the contrary, an amount of free air still present in the chamber prevents the pressure peak and delays its pressurization; hence, the mean resultant force tends to decrease the displacement (Figure 23c). The consequence when the pump works with the direct acting control (valve ON) is that, in the former case the pressure setting will be higher than the theoretical value [p.sub.1]*, while in the latter will be lower. For this reason, in the test with air the regulated pressure is significantly lower in comparison to the test without air. Instead when the piloted control is active (valve OFF), the displacement reduction is decided by the intervention of the valve RV. Even in this configuration, a small influence of the air content is observed, because of different working positions of the valve spool. In fact in presence of a negative component along the x-axis of the internal force, the stator equilibrium is reached with a higher value of the modulated pressure [p.sub.x] that implies a smaller opening of the valve, a smaller compression of its spring and finally a lower delivery pressure necessary to maintain in hydraulic equilibrium the spool. In this context, the time constants for aeration and dissolution play a fundamental role, but only within a specific range and in some operating conditions.

In Figure 24, the mean value and the amplitude of the simulated delivery pressure are plotted as function of the dissolution time constant at 5000 rpm and with aeration constant equal to [10.sup.-3] s.

In the configuration <<no air>>, if the dissolution constant is lower than [10.sup.-3] s, the air separated in the inlet volume has time enough to become again dissolved before the opening of the outlet port; in this case, a pressure peak is generated leading to the situation of Figure 23b. As the time constant increases, the completion of the dissolution process is delayed, with a progressive reduction of the pressure peak.

For values higher than about [10.sup.-2]-[10.sup.-1] s, the connection opens before the formation of the peak. The same trend is observed for the pressure amplitude; in this case, the increment of the time constant increases the amount of free air in the delivery line, leading to a dampening of the pressure ripple. In the configuration "with air", the dissolution time constant has no effect on the mean delivery pressure. In fact, the amount of separated air coming directly from the reservoir prevents the peak in the chamber regardless of the additional fraction of air released by the reduction of the pressure in the suction volume.

As far as the pressure amplitude is concerned, a very small time constant leads to the complete dissolution of the air in the oil in the delivery line. The consequence is that the back flow from the delivery volume due to the incomplete filling of the chamber generates high-pressure oscillations, being the equivalent bulk modulus coincident with the value of the pure oil (very stiff system). On the contrary, the bulk modulus is decreased by the presence of free air in the delivery line as the time constant increases.

In Figure 25, the analysis is repeated as function of the aeration time constant with constant dissolution time equal to 5 s.

In the configuration "no air", if the time constant is high there is no time enough to have the separation of a significant amount of air. In such a case, the bulk modulus remains high, the x-component of the internal force is positive due to the generation of the peak in the chambers and the delivery line is stiff.

For values of the time constant lower than [10.sup.-1] s, a fraction of the air becomes separated in the low-pressure volumes and due to the high dissolution time constant it remains undissolved also in the high-pressure volumes. The consequence is the progressive reduction of the pressure peak and the dampening of the oscillations in the delivery line. Times constants lower than [10.sup.-3] s are enough to cause the separation of the entire amount of air.

In the configuration "with air", the additional amount of air that can be released in the suction side does not alter the equivalent bulk modulus, which is maintained low by the amount of air coming directly from the reservoir.

Conclusions

A complete lumped parameter model of a variable displacement lubricating pump with the dynamics of the air release/dissolution processes has been presented. The model takes into account all leakage passageways and the forces acting on the stator ring. The validation on a simplified straight-pipe test circuit has allowed assessing the accuracy of the model. Thanks to the measurement of the amount of undissolved air, a crucial variable for the model tuning, normally unknown, has been eliminated. As far as the amount of dissolved air is concerned, the hypothesis of saturation condition has been adopted; however, the influence of a lower air percentage has been also assessed.

The implementation of the gas dynamics has brought to evidence the strong importance of the time constants; nevertheless, it must be highlighted that the operating parameters are influenced by the order of magnitude of the time constants and mostly within the range between [10.sup.-3] s and [10.sup.-1] s. The reason must be found in the time that the oil takes to cross the machine with the typical angular speeds of the hydraulic pumps. The assumptions made in the paper about the time for dissolution of the order of seconds and for aeration of the order of milliseconds, are in line with the limited data available in the open literature. As far as the pump behavior is concerned, the displacement control is more sensitive to separated air in the operating conditions of low-pressure setting. However, instability could occur when the pump works in the high-pressure (remote pilot) mode.

Based on the outcomes of the validation procedure, the present model has been proven to be a reliable tool for the design of the pump and of its control. It is straightforward that the model can be further improved by means of a tuning through CFD simulations, in order to determine precise values for the discharge coefficients in the resistive components.

Nomenclature

a - Smaller surface of the stator ring

A - Bigger surface of the stator ring

[b.sub.c] - Gap width

[b.sub.v] - Vane width

[b.sub.v1], [b.sub.v2] - Vane width fractions on which the chamber pressure acts

c - Journal bearing radial clearance

[C.sub.d] - Discharge coefficient

[C.sub.l] - Discharge coefficient for turbulent flow in vane tip gap

dm - Gas mass flow rate

[dmx.sub.u] - Undissolved gas mass flow rate

e - Eccentricity between rotor and stator

[f.sub.p] - Pressure dependent friction coefficient

[f.sub.v] - Velocity dependent friction coefficient

[F.sub.c] - Centrifugal force on the vane

[F.sub.fa], [F.sub.ft] - Axial and tangential friction force vane-rotor

[F.sub.i] - Resultant force on the stator track due to the pressure

[F.sub.p] - Force on the stator due to the chamber pressure

[F.sub.pa], [F.sub.pt] - Axial and tangential pressure forces on the vane

[F.sub.Ra] - Reaction force vane-stator

[F.sub.R1],[F.sub.R2] - Reaction forces vane-rotor

h - Vane lift

[h.sub.c] - Gap height

[h.sub.0] - Auxiliary constant - see Equation 11

[h.sub.min] - Minimum distance of the vane tip from the stator track

[h.sub.th] - Theoretical vane lift (vane in contact)

[h.sub.x] - Gap height at the generic coordinate x

H - Axial height of the rotor/stator

[I.sub.1], [I.sub.2] - Auxiliary constants - see Equation 12

[l.sub.c] - Gap length

[l.sub.v] - Vane length

[m.sub.v] - Vane mass

[m.sub.air] - Molecular mass of the air

p - Pressure (general)

[p.sub.1]* - Lower pressure setting of the displacement control

[p.sub.2]* - Higher pressure setting of the displacement control

[p.sub.d] - Delivery pressure

[p.sub.j] - Pressure in the i-th chamber

[p.sub.r] - Pressure acting on the vane roots

[p.sub.sat] - Saturation pressure of air in oil

[p.sub.x] - Pressure modulated by the relief valve

[r.sub.v] - Vane tip radius

R - Universal gas constant

[R.sub.r] - Rotor radius

[R.sub.s] - Stator track radius

[R.sub.sh] - Shaft radius

Q - Volumetric oil flow rate (general)

[Q.sub.i] - Volumetric flow rate entering the control volume

[Q.sub.l1...5] - Oil leakages - see Figure 5

S - Flow area

[T.sub.0] - Reference fluid temperature

[v.sub.0] - Sliding velocity vane tip - stator track

V - Volume

x - Coordinate along the vane tip gap

[x.sub.g] - Total mass gas fraction

[x.sub.u] - Undissolved gas mass fraction

[x.sub.u,eq] - Equilibrium undissolved gas mass fraction

[alpha] - Bunsen coefficient

[beta] - Bulk modulus

[delta] - Angle identifying the vane contact point

[[delta].sub.x] - See Figure 6

[DELTA]p - Pressure drop

[epsilon] - Journal bearing eccentricity ratio

[mu] - Dynamic viscosity

[xi] - Angle identifying the direction of the force [F.sub.p]

p - Fluid density

[p.sub.l] - Density of the liquid phase

[tau] - Aeration/dissolution time constant

[phi] - Shaft angle

[omega] - Shaft angular velocity

Acknowledgments

This study is in partial fulfillment of a research contract between and the Department of Energy of the Politecnico di Torino and Pierburg Pump Technology. Authors would like to thank Matteo Gasperini and Gaia Volandri for their contribution in providing to the Politecnico di Torino the data used for the simulation and for the model validation. Authors would like to acknowledge the LMS Amesim development team, headed by Olivier Delechelle at Siemens Industry Software Lyon, for sharing the information necessary for the implementation of the fluid model into the customized libraries developed at the Politecnico di Torino.

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Massimo Rundo, Politecnico di Torino

Raffaele Squarcini, Pierburg Pump Technology

Francesca Furno, Siemens Industry Software

History

Received: 10 Jul 2017

Revised: 21 Nov 2017

Accepted: 27 Jan 2018

e-Available: 18 Apr 2018

Keywords

Vane pump, Lubricating pump, Air dynamics, LMS Amesim

Citation

Rundo, M., Squarcini, R., and Furno, F., "Modelling of a Variable Displacement Lubricating Pump with Air Dissolution Dynamics," SAE Int. J. Engines 11(2):2018, doi:10.4271/03-11-02-0008.
TABLE 1 Fluid properties at 80 [degrees]C.

Quantity            Value      Unit

Density                812     kg/[m.sup.3]
Dynamic viscosity       10.2   cP
Bulk modulus          1320     MPa

TABLE 2 Definition of the 12 configurations used in the tests.

No.   Configuration               No.   Configuration

1     1474 rpm - OFF - with air    7    1474 rpm - OFF - no air
2     2468 rpm - OFF - with air    8    2468 rpm - OFF - no air
3     4474 rpm - OFF - with air    9    4474 rpm - OFF - no air
4     1474 rpm - ON - with air    10    1474 rpm - ON - no air
5     2468 rpm - ON - with air    11    2468 rpm - ON - no air
6     4474 rpm - ON - with air    12    4474 rpm - ON - no air
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Author:Rundo, Massimo; Squarcini, Raffaele; Furno, Francesca
Publication:SAE International Journal of Engines
Article Type:Technical report
Date:Apr 1, 2018
Words:9919
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