在軸向柱塞泵氣蝕問題的分析【中文3940字】【中英文WORD】

在軸向柱塞泵氣蝕問題的分析【中文3940字】【中英文WORD】,中文3940字,中英文WORD,軸向,柱塞,氣蝕,問題,分析,中文,3940,中英文,WORD 外文中文翻譯：【中文3940字】 在軸向柱塞泵氣蝕問題的分析 本論文討論和分析了一個柱塞孔與配流盤限制在軸向柱塞泵的控制量設計。真空是由柱塞的運動量引起的，需要由流動補償，否則，低氣壓可能導致的氣蝕和曝氣。配流盤幾何的研究，可以優(yōu)化一些分析性的限制，以防止蒸氣壓以下的柱塞壓力。配流盤的端口和缸體腰形窗口之間重疊的地方，設計時要考慮的空蝕和曝氣。 關鍵詞：空蝕，優(yōu)化，配流盤，負脈沖壓力 1介紹 在水壓機等液壓元件中，空穴或氣穴意味著，在低壓區(qū)液壓液體會出有空腔或氣泡形成以及崩潰在高壓地區(qū)，這將導致噪聲，振動，這將會降低效率。空蝕對泵的使用是極為不利的，這是因為倒塌形成的沖擊波可能像炸彈一樣足以損壞元件。當其壓力過低或溫度過高時，液壓油會蒸發(fā)。在實踐中，許多方法大多用于處理這些問題，比如：（1）提高油箱中的液位高度，（2油箱加壓，（3）提高泵的進口壓力，（4）降低泵內(nèi)流體的溫度，（5）特意設計的柱塞泵本身，對其結(jié)構(gòu)進行優(yōu)化設計。 在液壓機設計中的氣蝕現(xiàn)象，許多研究成果已取得一定的成果。在柱塞泵中，氣蝕主要可以分為兩種類型：一是與困油現(xiàn)象有關（這種現(xiàn)象可通過適當?shù)脑O計配流盤來阻止困油現(xiàn)象的發(fā)生）和所觀察到的層上收縮或擴大后的流動通道(由于旋轉(zhuǎn)設計所造成的)。在這項研究中處理氣蝕和測量氣缸壓力之間的關系。Edge and Darling報道了關于軸向柱塞泵內(nèi)的氣缸壓力的實驗研究。其中包括流體勢效應和氣蝕在氣缸內(nèi)高速度和高負荷條件的預測。另一項研究概述了液壓流體影響進氣條件和汽蝕潛力的觀點。它表明,物理屬性(如蒸汽壓力、粘度、密度和體積彈性模量)對適當?shù)卦u估影響潤滑和氣蝕是至關重要的。一個相似的氣蝕模型在熱力學性質(zhì)液體和蒸汽基礎上的用來理解了基本的物理現(xiàn)象的質(zhì)量流量減少和波動產(chǎn)生影響的液壓工具和噴射系統(tǒng)。Dular et al開發(fā)了一套專業(yè)系統(tǒng)用它來監(jiān)測和控制的液壓機械和調(diào)查氣蝕的可能性通過使用運用計算流體動力學(CFD)工具。通過一個簡單的單翼配置在一個空化隧道，氣蝕侵蝕作用已經(jīng)被測量和驗證。Alpha它假定了嚴重侵蝕經(jīng)常是由于一個主要的空穴飛轉(zhuǎn)的漩渦重復的崩潰所產(chǎn)生的。然后,在汽蝕強度通過一套簡單流參數(shù)可能擴大: 上游速度 ,空腔長度和壓力。一個新的空蝕裝置,稱為漩渦汽蝕生成器,介紹了各種侵蝕情況。更多的先前的研究已經(jīng)被集中在閥板的設計,在軸向柱塞泵中活塞和泵壓動力學與空穴現(xiàn)象相關聯(lián)?？刂企w積的方法和瞬時流(泄漏)正在深刻地研究中。Berta et al采用有限體積的概念發(fā)展了一個數(shù)學模型，壓力平衡槽的形式已經(jīng)被效仿和氣態(tài)的汽蝕被認為是在一個簡化的方式。一種改進的模型已經(jīng)被提出且實驗驗證了其結(jié)果。該模型可以分析氣缸壓力和流量的漣漪影響壓力平衡槽的設計。四種不同的數(shù)值模型的重點是液壓液體的特點,考慮到空穴以不同的方法協(xié)助減少流量振蕩。柱塞泵發(fā)展的經(jīng)驗表明,優(yōu)化的空穴現(xiàn)象應當包括下列問題: 發(fā)生氣蝕和空氣釋放、泵聲學引起的噪聲誘導、最大振幅的壓力波動,轉(zhuǎn)動力矩進展等。然而,這項研究的目的是修改配流盤的設計來防止氣蝕造成侵蝕蒸汽或空氣泡沫崩潰的墻壁上的軸向泵組件。與文學研究相反，這項研究主要集中在配流盤的幾何形狀和氣蝕分析之間的關系的發(fā)展。此優(yōu)化方法應用于分析的壓力脈沖與活塞孔內(nèi)飽和蒸汽壓。 2軸向柱塞泵,配流盤 典型軸向柱塞泵的原理如圖1所示。在這種情況下，軸偏移e的設計對降低成本是十分重要的。柱塞泵斜盤傾角度是可變的，決定每轉(zhuǎn)的排量即決定柱塞泵的流量 。如圖一所示，第N個柱塞滑靴組件轉(zhuǎn)過的轉(zhuǎn)角為在第n個柱塞滑靴組件沿x軸的位移可以寫成 xn = R tan（）sin（）+ a sec（） + e tan() (1) 其中R為柱塞滑靴組件的分布圓半徑。 此刻，在第n個柱塞的瞬時速度是 x˙n = R sin（）+ R tan（）cos（）+ R sin（） + e (2) 其中軸泵的旋轉(zhuǎn)速度=d / dt. 配流盤是約束柱塞泵流量的最重要的設備。配流盤吸排油窗口的幾何形狀以及瞬時相對缸體腰形窗口的位置通常被稱為配流盤的時間效應。配流盤開口與缸體底部的腰形窗口的重疊構(gòu)建流程區(qū)域或通道,它限制了柱塞泵的流體動力學，影響著其力學性能。在圖2中，在配流盤上吸排油窗口的角度分別為，.缸體配流腰形窗口的開啟角度為。在某些設計,在缸體腰形窗口和吸排油窗口的相對位置在上死點或下死點出現(xiàn)疊區(qū)域被稱為“cross-porting“在泵設計工程。 正是由于它的存在，使柱塞泵的容積效率大大的降低。被困量設計與交叉移植設計相比，它可以實現(xiàn)更好提高容積效率。然而, 在實踐中，交叉移植設計是通常用于改善噪音問題和泵穩(wěn)定問題。 3，柱塞腔有效容積的控制 在柱塞泵中，液體在活塞，缸體柱塞孔，滑靴，配流盤和斜板所組成的空間中流動如圖3所示。在每個柱塞空中，其瞬時質(zhì)量計算式為 = (3) 對上式求導可得 (4) 根據(jù)連續(xù)性方程，控制體積的質(zhì)量率為 (5) 其中為一個柱塞空中的瞬時流量 從體積彈性模量的定義可知， (6) 其中，Pn是柱塞孔內(nèi)的瞬時壓力。聯(lián)立方程（4）、（5）、（6）可得： (7) 其中泵軸旋轉(zhuǎn)角速度。 一個柱塞孔的瞬時流量，可以有計算式（1）求的: = + [R tan（）sin（）+ a sec（） + e tan() ] (8) 其中:為柱塞的有效截面積，是每個柱塞的有效容積 容積率的變化可以在一定的斜盤角度計算，即 =0，因此 (9) 聯(lián)立方程（7）、（8）、（9）可得 (10) 4、優(yōu)化設計 為了找到壓力超超量以及負脈沖可以用方程（10）進行優(yōu)化。 對一個非線性函數(shù)，為使其達到最大值和最小值通常是最優(yōu)化的目標。如果是一個閉區(qū)間上連續(xù)函數(shù)，其最大值和最小值必然存在。此外，函數(shù)的最大（或最?。?，要么必須是當?shù)刈畲螅ɑ蜃钚。┯虻膬?nèi)部或必須位于域的邊界上。因此，找到一個函數(shù)最大值的方法（或最小）是在所有的局部極大（或極?。┰趦?nèi)部檢測，評估在邊界上的極大（或極小）點，并選擇最大（或最?。Ｔ谥椎目刂企w積的壓力可能會發(fā)現(xiàn)，無論是作為最小或最大值為DP / DT = 0。因此，讓方程（10）的左邊等于零，可得： (11) 因此，柱塞泵吸油窗口的壓力不能太低，否則易產(chǎn)生汽蝕現(xiàn)象。在柱塞泵中的流量可能會通過在圖中表現(xiàn)出了幾個方案：（一）配流盤與缸體的間隙，（二）柱塞和缸體柱塞孔之間的間隙，（三）柱塞與滑靴之間的間隙，（四）滑靴與斜盤之間的間隙，（五）缸體底部腰形窗口與配流盤之間的重疊區(qū)域之間的間隙。由于泵的運行平穩(wěn)，在層流流動，可以計算為 (12) 其中為間隙的高度，為間隙的或通道的長度，其它參數(shù)具體參考流體力學相關知識。 流量對于（五），主要是在高速的情況下運轉(zhuǎn)，可以通過紊流方程來描述。 (14) 其中Pi和分別為柱塞泵吸排油窗口的壓力和，分別為每缸體腰形窗口和配流盤吸排油窗口單獨的瞬時之間的重疊面積。 非線性函數(shù)的旋轉(zhuǎn)角度面積的定義，這是由缸體腰形窗口，配流盤吸排油窗口，阻尼槽，減壓孔，等等幾何圖形決定的。 結(jié)合方程（11）、（12）、（13）、(14)，則此面積的方程為： (15) 其中為總重疊面積，= 且的定義為： 在柱塞孔中，壓力從低到高不等，而通過配流盤吸油口與排油口。這可能是瞬時壓力達到極低值在吸油窗口（，圖2所示），這有可能使其低于大氣壓，即;此時柱塞泵非常容易發(fā)生汽蝕，為阻止此類現(xiàn)象的發(fā)生，總的重疊量的面積設計應滿足 (16) 其中的最小面積為= 的定義為： 蒸氣壓的壓力下，液體蒸發(fā)成氣態(tài)形式。根據(jù)克勞修斯 - 克拉珀龍有關的任何物質(zhì)的蒸氣壓隨溫度的非線性。隨著溫度的逐步增加，蒸汽壓力變得足以克服顆粒的吸引力和內(nèi)部的物質(zhì)，使液體形成氣泡。對于純介質(zhì)，蒸氣壓可以通過溫度用Antoine方程來確定，，其中，T為油液溫度，A,B,C分別為常數(shù)。 當柱塞經(jīng)過配流盤吸油口時，壓力變化依賴于余弦函數(shù)式（10）。據(jù)悉，有一些典型的柱塞位置與配流盤的進油口，重疊的開頭和結(jié)尾等有關，TDC和BDC（）和零排量位置（ =0）。將討論這兩種情況如下： （1） 當時，它未必總是要保持重疊量，因為滑流可提供流量來填補真空的重疊區(qū)域。從方程（16）可知，讓=0, 上下死點的時間角度可設計為 (17) （2）當 = 0時，cos函數(shù)有最大值，它可以提供另一個重疊區(qū)域的限制，以防止負壓力脈沖，比如： (18) 其中，為最小重疊區(qū)域 為了防止低壓腔柱塞出現(xiàn)氣泡，方程（16）中蒸氣壓壓力設置較低的限制。那么整體的重疊區(qū)域，可以得出有一個設計上的極限。此限制確定條件下的泄漏，蒸汽壓力，轉(zhuǎn)速等決定。它表明柱塞泵轉(zhuǎn)速越高，越可能發(fā)生更嚴重的氣蝕，因此設計需要讓更多的重疊區(qū)域在柱塞孔中。在另一邊，低蒸氣壓力的液壓油是首選減少的機會，以達到空化條件。事實上，空氣釋放開始在更高的壓力主要集中在純剪切湍流層蝕過程中，發(fā)生在場景五。因此，如果存在大量被困，并溶解在液體中的空氣，蒸汽壓力可能得適應重疊區(qū)域的設計在方程（16）中。 通過上述間隙層滲漏是一個在設計上的權衡。它表明，從柱塞越多的泄露可能會減輕氣蝕問題。然而，越多的泄漏，可能會降低泵的效率在排油窗口。在一些設計的情況下，最大定時角可以由方程（17）決定，不可兼得的在TDC和BDC的同時擁有大的重疊的和非常低的壓力。 6結(jié)論 配流盤的設計是一個關鍵問題在解決柱塞泵發(fā)生氣蝕現(xiàn)象的設計中。本研究采用控制體積法來分析一個柱塞內(nèi)承擔相關的流量，壓力和泄漏的配流盤計時。如果重疊區(qū)域由缸體腰形窗口和配流盤的開口開發(fā)設計不當，就不會有足夠的流量來補充由于旋轉(zhuǎn)運動引起的空間。因此，活塞的壓力可能會低于飽和蒸汽壓力，易形成蒸汽氣泡。為了控制有害的氣泡，優(yōu)化方法用于檢測通過配流盤時間限制的。分析重疊區(qū)域限制的最低壓力，需要得到滿足，以保持壓力，不會有大的負脈沖壓力導致系統(tǒng)在很大程度上增強汽蝕問題。 在這項研究中，柱塞控制量的動態(tài)開發(fā)利用恒定的流量系數(shù)和層滲漏等幾個假設。實際上是非線性的基礎上的幾何形狀，流量參量等，以及在實踐中，由于振動和動力漣漪，特意控制泄漏間隙，流量系數(shù)可能不會保持恒定的高度和寬度。所有這些問題大多數(shù)情況下是根據(jù)大量的經(jīng)驗考慮的的，是一個個非常錯綜復雜問題以及需要進一步的研究。本文給出的結(jié)果可以更準確地估計這些研究提供方便。 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