【文章內(nèi)容簡介】 ption pressure to cause cavitations. The term of ta n( ) cos( )pnAR? ? ? in Eq. (11) has the positive value in the range of intake ports ( 22???? ? ? ), 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 nq 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] 312IV kkLn iIkhqpL???? ? (12) where kh is the height of the clearance, kL is the passage length, scenarios I–IV mostly have low Reynolds numbers and can be regarded k? is the width of the clearance (note that in the scenario II, k? =2? r, in which r is the piston radius), and p is the pressure drop defined in the intake ports as p = cp np (13) where cp 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 plicated 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, ., minimum gap flows. For scenario V, the flow is mostly in high velocity and can be described by using the turbulent orifice equation as 2 ( ) 2 ( )( ) ( )i n d nT n d i d dp p P pq c A c A???????? (14) where Pi and Pd are the intake and discharge pressure of the pump and ()iA? and ()dA? 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, depression holes, and so forth. Combining Eqs. (11) –(14), the area can be obtained as 3ta n( ) c os( ) ( )12()2( )K I V kkP n n ckIkd i nhA r p pLAc p p?? ? ?? ??????? ??(15) where ()A? is the total overlap area of ()A? = ( ) ( )idAA? ? ?? , and ? is defined as d n i np p p p?? ? ?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( 22???? ? ? shown in Fig. 2) that may be located below the vapor pressure vpp , ., n vppp? 。then cavitations can happen. To prevent the phenomena, the total overlap area of ()A? might be designed to be satisfied with 30ta n ( ) c o s( ) ( )12() 2 ( )K I V kkP n v p ckI kd i v phA r p pLAc p p?? ? ?? ??????? ??(16) where 0()A? is the minimum area of 0()A? = 0( ) ( )idAA? ? ?? and 0? is a constant that is 0 /d v p i v pp p p p? ? ? ? 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 bees sufficient to overe particle attraction and make the liquid form bubbles inside the substance. For pure ponents, the vapor pressure can be determined by the temperature using the Antoine equation as /( )10A B C T??, 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, ., TDC and BDC ( / 2, / 2? ? ??? ) and the zero displacement position (? =0). The two situations will be discussed as follows: (1) When / 2, / 2? ? ??? , it is not always necessary to maintain the overlap area of 0()A? because slip flows may provide filling up for the vacuum. From Eq. (16), letting 0()A? =0, the timing angles at the TDC and BDC may be designed as 31c o s ( )ta n ( ) 1 2 2IVc v p kki IPkpp hA r L? ?? ? ? ?? ???? (17) in which the open angle of the barrel kidney is . There is no crossporting 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 that30ta n ( ) ( )12( 0 ) 2 ( )K I V kkP v p ckI kd i