外文原文： The Analysis of Cavitation Problems in the Axial Piston Pump shu Wang Eaton Corporation, 14615 Lone Oak Road, Eden Prairie, MN 55344 This paper discusses and analyzes the control volume of a piston bore constrained by the valve plate in axial piston pumps. The vacuum within the piston bore caused by the rise volume needs to be compensated by the flow; otherwise, the low pressure may cause the cavitations and aerations. In the research, the valve plate geometry can be optimized by some analytical limitations to prevent the piston pressure below the vapor pressure. The limitations provide the design guide of the timings and overlap areas between valve plate ports and barrel kidneys to consider the cavitations and aerations. _DOI: 10.1115/1.4002058_ Keywords: cavitation , optimization, valve plate, pressure undershoots 1 Introduction In hydrostatic machines, cavitations mean that cavities or bubbles form in the hydraulic liquid at the low pressure and collapse at the high pressure region, which causes noise, vibration, and less efficiency. Cavitations are undesirable in the pump since the shock waves formed by collapsed may be strong enough to damage components. The hydraulic fluid will vaporize when its pressure becomes too low or when the temperature is too high. In practice, a number of approaches are mostly used to deal with the problems: (1) raise the liquid level in the tank, (2) pressurize the tank, (3) booster the inlet pressure of the pump, (4) lower the pumping fluid temperature, and (5) design deliberately the pump itself. Many research efforts have been made on cavitation phenomena in hydraulic machine designs. The cavitation is classified into two types in piston pumps: trapping phenomenon related one (which can be prevented by the proper design of the valve plate) and the one observed on the layers after the contraction or enlargement of flow passages (caused by rotating group designs) in Ref. (1). The relationship between the cavitation and the measured cylinder pressure is addressed in this study. Edge and Darling (2) reported an experimental study of the cylinder pressure within an axial piston pump. The inclusion of fluid momentum effects and cavitations within the cylinder bore are predicted at both high speed and high load conditions. Another study in Ref. (3) provides an overview of hydraulic fluid impacting on the inlet condition and cavitation potential. It indicates that physical properties (such as vapor pressure, viscosity, density, and bulk modulus) are vital to properly evaluate the effects on lubrication and cavitation. A homogeneous cavitation model based on the thermodynamic properties of the liquid and steam is used to understand the basic physical phenomena of mass flow reduction and wave motion influences in the hydraulic tools and injection systems (4). Dular et al. (5, 6) developed an expert system for monitoring and control of cavitations in hydraulic machines and investigated the possibility of cavitation erosion by using the computational fluid dynamics (CFD) tools. The erosion effects of cavitations have been measured and validated by a simple single hydrofoil configuration in a cavitation tunnel. It is assumed that the severe erosion is often due to the repeated collapse of the traveling vortex generated by a leading edge cavity in Ref. (7). Then, the cavitation erosion intensity may be scaled by a simple set of flow parameters: the upstream velocity, the Strouhal number, the cavity length, and the pressure. A new cavitation erosion device, called vortex cavitation generator, is introduced to comparatively study various erosion situations (8). More previous research has been concentrated on the valve plate designs, piston, and pump pressure dynamics that can be associated with cavitations in axial piston pumps. The control volume approach and instantaneous flows (leakage) are profoundly studied in Ref. [9]. Berta et al. [10] used the finite volume concept to develop a mathematical model in which the effects of port plate relief grooves have been modeled and the gaseous cavitation is considered in a simplified manner. An improved model is proposed in Ref. [11] and validated by experimental results. The model may analyze the cylinder pressure and flow ripples influenced by port plate and relief groove design. Manring compared principal advantages of various valve plate slots (i.e., the slots with constant, linearly varying, and quadratic varying areas) in axial piston pumps [12]. Four different numerical models are focused on the characteristics of hydraulic fluid, and cavitations are taken into account in different ways to assist the reduction in flow oscillations [13]. The experiences of piston pump developments show that the optimization of the cavitations/aerations shall include the following issues: occurring cavitation and air release, pump acoustics caused by the induced noises, maximal amplitudes of pressure fluctuations, rotational torque progression, etc. However, the aim of this study is to modify the valve plate design to prevent cavitation erosions caused by collapsing steam or air bubbles on the walls of axial pump components. In contrast to literature studies, the research focuses on the development of analytical relationship between the valve plate geometrics and cavitations. The optimization method is applied to analyze the pressure undershoots compared with the saturated vapor pressure within the piston bore. The appropriate design of instantaneous flow areas between the valve plate and barrel kidney can be decided consequently. 2 The Axial Piston Pump and Valve Plate The typical schematic of the design of the axis piston pump is shown in Fig. 1. The shaft offset e is designed in this case to generate stroking containment moments for reducing cost purposes. The variation between the pivot center of the slipper and swash rotating center is shown as a. The swash angle is the variable that determines the amount of fluid pumped per shaft revolution. In Fig. 1, the nth piston-slipper assembly is located at the angle of . The displacement of the nth piston-slipper assembly along the x-axis can be written as xn = R tan（）sin（）+ a sec（） + e tan() (1) where R is the pitch radius of the rotating group. Then, the instantaneous velocity of the nth piston is x˙n = R sin（）+ R tan（）cos（）+ R sin（） + e (2) where the shaft rotating speed of the pump is=d / dt. The valve plate is the most significant device to constraint flow in piston pumps. The geometry of intake/discharge ports on the valve plate and its instantaneous relative positions with respect to barrel kidneys are usually referred to the valve plate timing. The ports of the valve plate overlap with each barrel kidneys to construct a flow area or passage, which confines the fluid dynamics of the pump. In Fig. 2, the timing angles of the discharge and intake ports on the valve plate are listed as and . The opening angle of the barrel kidney is referred to as . In some designs, there exists a simultaneous overlap between the barrel kidney and intake/discharge slots at the locations of the top dead center (TDC) or bottom dead center (BDC) on the valve plate on which the overlap area appears together referred to as “cross-porting” in the pump design engineering. The cross-porting communicates the discharge and intake ports, which may usually lower the volumetric efficiency. The trapped-volume design is compared with the design of the cross-porting, and it can achieve better efficiency 14]. However, the cross-porting is Fig. 1 The typical axis piston pump commonly used to benefit the noise issue and pump stability in practice. 3 The Control Volume of a Piston Bore In the piston pump, the fluid within one piston is embraced by the piston bore, cylinder barrel, slipper, valve plate, and swash plate shown in Fig. 3. There exist some types of slip flow by virtue of relative Fig. 2 Timing of the valve plate motions and clearances between thos e components. Within the control volume of each piston bore, the instantaneous mass is calculated as = (3) where and are the instantaneous density and volume such that the mass time rate of change can be given as Fig. 3 The control volume of the piston bore (4) where d is the varying of the volume. Based on the conservation equation, the mass rate in the control volume is (5) where is the instantaneous flow rate in and out of one piston. From the definition of the bulk modulus, (6) where Pn is the instantaneous pressure within the piston bore. Substituting Eqs. (5) and (6) into Eq. (4) yields (7) where the shaft speed of the pump is . The instantaneous volume of one piston bore can be calculated by using Eq. (1) as = + [R tan（）sin（）+ a sec（） + e tan() ] (8) where is the piston sectional area and is the volume of each piston, which has zero displacement along the x-axis (when =0, ). The volume rate of change can be calculated at the certain swash angle, i.e., =0, such that (9) in which it is noted that the piston bore volume increases or decreases with respect to the rotating angle of . Substituting Eqs. (8) and (9) into Eq. (7) yields (10) 4 Optimal Designs To find the extrema of pressure overshoots and undershoots in the control volume of piston bores, the optimization method can be used in Eq. (10). In a nonlinear function, reaching global maxima and minima is usually the goal of optimization. If the function is continuous on a closed interval, global maxima and minima exist. Furthermore, the global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain or must lie on the boundary of the domain. So, the method of finding a global maximum (or minimum) is to detect all the local maxima (or minima) in the interior, evaluate the maxima (or minima) points on the boundary, and select the biggest (or smallest) one. Local maximum or local minimum can be searched by using the first derivative test that the potential extrema of a function f( · ), with derivative , can solve the equation at the critical points of =0 [15]. The pressure of control volumes in the piston bore may be found as either a minimum or maximum value as dP/ dt=0. Thus, letting the left side of Eq. (10) be equal to zero yields (11) In a piston bore, the quantity of offsets the volume varying and then decreases the overshoots and undershoots of the piston pressure. In this study, the most interesting are undershoots of the pressure, which may fall below the vapor pressure or gas desorption pressure to cause cavitations. The term of in Eq. (11) has the positive value in the range of intake ports (), shown in Fig. 2, which means that the piston volume arises. Therefore, the piston needs the sufficient flow in; otherwise, the pressure may drop. In the piston, the flow of may get through in a few scenarios shown in Fig. 3: (I) the clearance between the valve plate and cylinder barrel, (II) the clearance between the cylinder bore and piston, (III) the clearance between the piston and slipper, (IV) the clearance between the slipper and swash plate, and (V) the overlapping area between the barrel kidney and valve plate ports. As pumps operate stably, the flows in the as laminar flows, which can be calculated as
[16] (12) where is the height of the clearance, is the passage length, scenarios I–IV mostly have low Reynolds numbers and can be regarded is the width of the clearance (note that in the scenario II, =2· r, in which r is the piston radius), and is the pressure drop defined in the intake ports as =- (13) where is the case pressure of the pump. The fluid films through the above clearances were extensively investigated in previous research. The effects of the main related dimensions of pump and the operating conditions on the film are numerically clarified in Refs.
[17,18]. The dynamic behavior of slipper pads and the clearance between the slipper and swash plate can be referred to Refs.
[19,20]. Manring et al.
[21,22] investigated the flow rate and load carrying capacity of the slipper bearing in theoretical and experimental methods under different deformation conditions. A simulation tool called CASPAR is used to estimate the nonisothermal gap flow between the cylinder barrel and the valve plate by Huang and Ivantysynova
[23]. The simulation program also considers the surface deformations to predict gap heights, frictions, etc., between the piston and barrel and between the swash plate and slipper. All these clearance geometrics in Eq. (12) are nonlinear and operation based, which is a complicated issue. In this study, the experimental measurements of the gap flows are preferred. If it is not possible, the worst cases of the geometrics or tolerances with empirical adjustments may be used to consider the cavitation issue, i.e., minimum gap flows. For scenario V, the flow is mostly in high velocity and can be described by using the turbulent orifice equation as (14) where Pi and Pd are the intake and discharge pressure of the pump and and are the instantaneous overlap area between barrel kidneys and inlet/discharge ports of the valve plate individually. The areas are nonlinear functions of the rotating angle, which is defined by the geometrics of the barrel kidney, valve plate ports, silencing grooves, decompression holes, and so forth. Combining Eqs. (11) –(14), the area can be obtained as (15) where is the total overlap area of =, and is defined as In the piston bore, the pressure varies from low to high while passing over the intake and discharge ports of the valve plates. It is possible that the instantaneous pressure achieves extremely low values during the intake area( shown in Fig. 2) that may be located below the vapor pressure , i.e., ;then cavitations can happen. To prevent the phenomena, the total overlap area of might be designed to be satisfied with (16) where is the minimum area of = and is a constant that is Vapor pressure is the pressure under which the liquid evaporates into a gaseous form. The vapor pressure of any substance increases nonlinearly with temperature according to the Clausius–Clapeyron relation. With the incremental increase in temperature, the vapor pressure becomes sufficient to overcome particle attraction and make the liquid form bubbles inside the substance. For pure components, the vapor pressure can be determined by the temperature using the Antoine equation as , where T is the temperature, and A, B, and C are constants
[24]. As a piston traverse the intake port, the pressure varies dependent on the cosine function in Eq. (10). It is noted that there are some typical positions of the piston with respect to the intake port, the beginning and ending of overlap, i.e., TDC and BDC ( ) and the zero displacement position ( =0). The two situations will be discussed as follows: (1) When, it is not always necessary to maintain the overlap area of because slip flows may provide filling up for the vacuum. From Eq. (16), letting =0, the timing angles at the TDC and BDC may be designed as (17) in which the open angle of the barrel kidney is . There is no cross-porting flow with the timing in the intake port. (2) When =0, the function of cos has the maximum value, which can provide another limitation of the overlap area to prevent the low pressure undershoots such that (18) where is the minimum overlap area of . To prevent the low piston pressure building bubbles, the vapor pressure is considered as the lower limitation for the pressure settings in Eq. (16). The overall of overlap areas then can be derived to have a design limitation. The limitation is determined by the leakage conditions, vapor pressure, rotating speed, etc. It indicates that the higher the pumping speed, the more severe cavitation may happen, and then the designs need more overlap area to let flow in the piston bore. On the other side, the low vapor pressure of the hydraulic fluid is preferred to reduce the opportunities to reach the cavitation conditions. As a result, only the vapor pressure of the pure fluid is considered in Eqs. (16)–(18). In fact, air release starts in the higher pressure than the pure cavitation process mainly in turbulent shear layers, which occur in scenario V. Therefore, the vapor pressure might be adjusted to design the overlap area by Eq. (16) if there exists substantial trapped and dissolved air in the fluid. The laminar leakages through the clearances aforementioned are a tradeoff in the design. It is demonstrated that the more leakage from the pump case to piston may relieve cavitation problems.However, the more leakage may degrade the pump efficiency in the discharge ports. In some design cases, the maximum timing angles can be determined by Eq. (17)to not have both simultaneous overlapping and highly low pressure at the TDC and BDC. While the piston rotates to have the zero displacement, the minimum overlap area can be determined by Eq. 18, which may assist the piston not to have the large pressure undershoots during flow intake. 6 Conclusions The valve plate design is a critical issue in addressing the cavitation or aeration phenomena in the piston pump. This study uses the control volume method to analyze the flow, pressure, and leakages within one piston bore related to the valve plate timings. If the overlap area developed by barrel kidneys and valve plate ports is not properly designed, no sufficient flow replenishes the rise volume by the rotating movement. Therefore, the piston pressure may drop below the saturated vapor pressure of the liquid and air ingress to form the vapor bubbles. To control the damaging cavitations, the optimization approach is used to detect the lowest pressure constricted by valve plate timings. The analytical limitation of the overlap area needs to be satisfied to remain the pressure to not have large undershoots so that the system can be largely enhanced on cavitation/aeration issues. In this study, the dynamics of the piston control volume is developed by using several assumptions such as constant discharge coefficients and laminar leakages. The discharge coefficient is practically nonlinear based on the geometrics, flow number, etc. Leakage clearances of the control volume may not keep the constant height and width as well in practice due to vibrations and dynamical ripples. All these issues are complicated and very empirical and need further consideration in the future. The results presented in this paper can be more accurate in estimating the cavitations with these extensive studies. Nomenclature the total overlap area between valve plate ports and barrel kidneys Ap = piston section area A, B, C= constants A= offset between the piston-slipper joint and surface of the swash plate = orifice discharge coefficient e= offset between the swash plate pivot and the shaft centerline of the pump = the height of the clearance = the passage length of the clearance M= mass of the fluid within a single piston (kg) N= number of pistons n = piston and slipper counter = fluid pressure and pressure drop (bar) Pc= the case pressure of the pump (bar) Pd= pump discharge pressure (bar) Pi = pump intake pressure (bar) Pn = fluid pressure within the nth piston bore (bar) Pvp = the vapor pressure of the hydraulic fluid(bar) qn, qLn, qTn = the instantaneous flow rate of each piston