在轴向柱塞泵气蚀问题的分析【中文3940字】【中英文WORD】
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外文中文翻译：【中文3940字】
在轴向柱塞泵气蚀问题的分析
本论文讨论和分析了一个柱塞孔与配流盘限制在轴向柱塞泵的控制量设计。真空是由柱塞的运动量引起的，需要由流动补偿，否则，低气压可能导致的气蚀和曝气。配流盘几何的研究，可以优化一些分析性的限制，以防止蒸气压以下的柱塞压力。配流盘的端口和缸体腰形窗口之间重叠的地方，设计时要考虑的空蚀和曝气。
关键词：空蚀，优化，配流盘，负脉冲压力
1介绍
在水压机等液压元件中，空穴或气穴意味着，在低压区液压液体会出有空腔或气泡形成以及崩溃在高压地区，这将导致噪声，振动，这将会降低效率。空蚀对泵的使用是极为不利的，这是因为倒塌形成的冲击波可能像炸弹一样足以损坏元件。当其压力过低或温度过高时，液压油会蒸发。在实践中，许多方法大多用于处理这些问题，比如：（1）提高油箱中的液位高度，（2油箱加压，（3）提高泵的进口压力，（4）降低泵内流体的温度，（5）特意设计的柱塞泵本身，对其结构进行优化设计。
在液压机设计中的气蚀现象，许多研究成果已取得一定的成果。在柱塞泵中，气蚀主要可以分为两种类型：一是与困油现象有关（这种现象可通过适当的设计配流盘来阻止困油现象的发生）和所观察到的层上收缩或扩大后的流动通道(由于旋转设计所造成的)。在这项研究中处理气蚀和测量气缸压力之间的关系。Edge and Darling报道了关于轴向柱塞泵内的气缸压力的实验研究。其中包括流体势效应和气蚀在气缸内高速度和高负荷条件的预测。另一项研究概述了液压流体影响进气条件和汽蚀潜力的观点。它表明,物理属性(如蒸汽压力、粘度、密度和体积弹性模量)对适当地评估影响润滑和气蚀是至关重要的。一个相似的气蚀模型在热力学性质液体和蒸汽基础上的用来理解了基本的物理现象的质量流量减少和波动产生影响的液压工具和喷射系统。Dular et al开发了一套专业系统用它来监测和控制的液压机械和调查气蚀的可能性通过使用运用计算流体动力学(CFD)工具。通过一个简单的单翼配置在一个空化隧道，气蚀侵蚀作用已经被测量和验证。Alpha它假定了严重侵蚀经常是由于一个主要的空穴飞转的漩涡重复的崩溃所产生的。然后,在汽蚀强度通过一套简单流参数可能扩大: 上游速度 ,空腔长度和压力。一个新的空蚀装置,称为漩涡汽蚀生成器,介绍了各种侵蚀情况。更多的先前的研究已经被集中在阀板的设计,在轴向柱塞泵中活塞和泵压动力学与空穴现象相关联。控制体积的方法和瞬时流(泄漏)正在深刻地研究中。Berta et al采用有限体积的概念发展了一个数学模型，压力平衡槽的形式已经被效仿和气态的汽蚀被认为是在一个简化的方式。一种改进的模型已经被提出且实验验证了其结果。该模型可以分析气缸压力和流量的涟漪影响压力平衡槽的设计。四种不同的数值模型的重点是液压液体的特点,考虑到空穴以不同的方法协助减少流量振荡。柱塞泵发展的经验表明,优化的空穴现象应当包括下列问题: 发生气蚀和空气释放、泵声学引起的噪声诱导、最大振幅的压力波动,转动力矩进展等。然而,这项研究的目的是修改配流盘的设计来防止气蚀造成侵蚀蒸汽或空气泡沫崩溃的墙壁上的轴向泵组件。与文学研究相反，这项研究主要集中在配流盘的几何形状和气蚀分析之间的关系的发展。此优化方法应用于分析的压力脉冲与活塞孔内饱和蒸汽压。
2轴向柱塞泵,配流盘
典型轴向柱塞泵的原理如图1所示。在这种情况下，轴偏移e的设计对降低成本是十分重要的。柱塞泵斜盘倾角度是可变的，决定每转的排量即决定柱塞泵的流量 。如图一所示，第N个柱塞滑靴组件转过的转角为在第n个柱塞滑靴组件沿x轴的位移可以写成
xn = R tan（）sin（）+ a sec（） + e tan() (1)
其中R为柱塞滑靴组件的分布圆半径。
此刻，在第n个柱塞的瞬时速度是
x˙n = R sin（）+ R tan（）cos（）+ R sin（） + e (2)
其中轴泵的旋转速度=d / dt.
配流盘是约束柱塞泵流量的最重要的设备。配流盘吸排油窗口的几何形状以及瞬时相对缸体腰形窗口的位置通常被称为配流盘的时间效应。配流盘开口与缸体底部的腰形窗口的重叠构建流程区域或通道,它限制了柱塞泵的流体动力学，影响着其力学性能。在图2中，在配流盘上吸排油窗口的角度分别为，.缸体配流腰形窗口的开启角度为。在某些设计,在缸体腰形窗口和吸排油窗口的相对位置在上死点或下死点出现叠区域被称为“cross-porting“在泵设计工程。
正是由于它的存在，使柱塞泵的容积效率大大的降低。被困量设计与交叉移植设计相比，它可以实现更好提高容积效率。然而, 在实践中，交叉移植设计是通常用于改善噪音问题和泵稳定问题。
3，柱塞腔有效容积的控制
在柱塞泵中，液体在活塞，缸体柱塞孔，滑靴，配流盘和斜板所组成的空间中流动如图3所示。在每个柱塞空中，其瞬时质量计算式为 = (3)
对上式求导可得
(4)
根据连续性方程，控制体积的质量率为
(5)
其中为一个柱塞空中的瞬时流量
从体积弹性模量的定义可知，
(6)
其中，Pn是柱塞孔内的瞬时压力。联立方程（4）、（5）、（6）可得：
(7)
其中泵轴旋转角速度。
一个柱塞孔的瞬时流量，可以有计算式（1）求的:
= + [R tan（）sin（）+ a sec（） + e tan() ] (8)
其中:为柱塞的有效截面积，是每个柱塞的有效容积
容积率的变化可以在一定的斜盘角度计算，即 =0，因此
(9)
联立方程（7）、（8）、（9）可得
(10)
4、优化设计
为了找到压力超超量以及负脉冲可以用方程（10）进行优化。
对一个非线性函数，为使其达到最大值和最小值通常是最优化的目标。如果是一个闭区间上连续函数，其最大值和最小值必然存在。此外，函数的最大（或最小），要么必须是当地最大（或最小）域的内部或必须位于域的边界上。因此，找到一个函数最大值的方法（或最小）是在所有的局部极大（或极小）在内部检测，评估在边界上的极大（或极小）点，并选择最大（或最小）。在柱塞孔的控制体积的压力可能会发现，无论是作为最小或最大值为DP / DT = 0。因此，让方程（10）的左边等于零，可得：
(11)
因此，柱塞泵吸油窗口的压力不能太低，否则易产生汽蚀现象。在柱塞泵中的流量可能会通过在图中表现出了几个方案：（一）配流盘与缸体的间隙，（二）柱塞和缸体柱塞孔之间的间隙，（三）柱塞与滑靴之间的间隙，（四）滑靴与斜盘之间的间隙，（五）缸体底部腰形窗口与配流盘之间的重叠区域之间的间隙。由于泵的运行平稳，在层流流动，可以计算为
(12)
其中为间隙的高度，为间隙的或通道的长度，其它参数具体参考流体力学相关知识。
流量对于（五），主要是在高速的情况下运转，可以通过紊流方程来描述。
(14)
其中Pi和分别为柱塞泵吸排油窗口的压力和，分别为每缸体腰形窗口和配流盘吸排油窗口单独的瞬时之间的重叠面积。
非线性函数的旋转角度面积的定义，这是由缸体腰形窗口，配流盘吸排油窗口，阻尼槽，减压孔，等等几何图形决定的。
结合方程（11）、（12）、（13）、(14)，则此面积的方程为：
(15)
其中为总重叠面积，=
且的定义为：
在柱塞孔中，压力从低到高不等，而通过配流盘吸油口与排油口。这可能是瞬时压力达到极低值在吸油窗口（，图2所示），这有可能使其低于大气压，即;此时柱塞泵非常容易发生汽蚀，为阻止此类现象的发生，总的重叠量的面积设计应满足
(16)
其中的最小面积为=
的定义为：
蒸气压的压力下，液体蒸发成气态形式。根据克劳修斯 - 克拉珀龙有关的任何物质的蒸气压随温度的非线性。随着温度的逐步增加，蒸汽压力变得足以克服颗粒的吸引力和内部的物质，使液体形成气泡。对于纯介质，蒸气压可以通过温度用Antoine方程来确定，，其中，T为油液温度，A,B,C分别为常数。
当柱塞经过配流盘吸油口时，压力变化依赖于余弦函数式（10）。据悉，有一些典型的柱塞位置与配流盘的进油口，重叠的开头和结尾等有关，TDC和BDC（）和零排量位置（ =0）。将讨论这两种情况如下：
（1） 当时，它未必总是要保持重叠量，因为滑流可提供流量来填补真空的重叠区域。从方程（16）可知，让=0,
上下死点的时间角度可设计为
(17)
（2）当 = 0时，cos函数有最大值，它可以提供另一个重叠区域的限制，以防止负压力脉冲，比如： (18)
其中，为最小重叠区域
为了防止低压腔柱塞出现气泡，方程（16）中蒸气压压力设置较低的限制。那么整体的重叠区域，可以得出有一个设计上的极限。此限制确定条件下的泄漏，蒸汽压力，转速等决定。它表明柱塞泵转速越高，越可能发生更严重的气蚀，因此设计需要让更多的重叠区域在柱塞孔中。在另一边，低蒸气压力的液压油是首选减少的机会，以达到空化条件。事实上，空气释放开始在更高的压力主要集中在纯剪切湍流层蚀过程中，发生在场景五。因此，如果存在大量被困，并溶解在液体中的空气，蒸汽压力可能得适应重叠区域的设计在方程（16）中。
通过上述间隙层渗漏是一个在设计上的权衡。它表明，从柱塞越多的泄露可能会减轻气蚀问题。然而，越多的泄漏，可能会降低泵的效率在排油窗口。在一些设计的情况下，最大定时角可以由方程（17）决定，不可兼得的在TDC和BDC的同时拥有大的重叠的和非常低的压力。
6结论
配流盘的设计是一个关键问题在解决柱塞泵发生气蚀现象的设计中。本研究采用控制体积法来分析一个柱塞内承担相关的流量，压力和泄漏的配流盘计时。如果重叠区域由缸体腰形窗口和配流盘的开口开发设计不当，就不会有足够的流量来补充由于旋转运动引起的空间。因此，活塞的压力可能会低于饱和蒸汽压力，易形成蒸汽气泡。为了控制有害的气泡，优化方法用于检测通过配流盘时间限制的。分析重叠区域限制的最低压力，需要得到满足，以保持压力，不会有大的负脉冲压力导致系统在很大程度上增强汽蚀问题。
在这项研究中，柱塞控制量的动态开发利用恒定的流量系数和层渗漏等几个假设。实际上是非线性的基础上的几何形状，流量参量等，以及在实践中，由于振动和动力涟漪，特意控制泄漏间隙，流量系数可能不会保持恒定的高度和宽度。所有这些问题大多数情况下是根据大量的经验考虑的的，是一个个非常错综复杂问题以及需要进一步的研究。本文给出的结果可以更准确地估计这些研究提供方便。
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.
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
(l/min)
R = piston pitch radius
r
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