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Purdue ME 59700 - Lecture notes

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Dr. Monika IvantysynovaMAHA Professor Fluid Power SystemsDesign and Modeling of Fluid Power SystemsME 597/ABE 591 - Lecture 7MAHA Fluid Power Research CenterPurdue UniversitySICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 5912Content of 6th lecture 2Aim:Derivation of basic equations for gap flow and calculation ofGap flow parameters (pressure and velocity distribution, load ability,gap flow and viscous friction)Basic knowledge about the gap design and simulation modelsThe lubricating gap as a basic design element of displacement machinesGap flow calculation. Gap with a constant gap heightGap with variable gap height (slipper)Gap between piston and cylinder, numerical solution of gap flow equationsNon-isothermal gap flowSICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 5913Lubricating Gap3Basic Design Element of Displacement Machinesp2xGaps in Displacement MachinesWith Variable Gap HeightpxhvDesign & operatingparameter Gap heightGap flowLoad abilityViscous frictionLeakagePerformance & Losses Sealing FunctionBearing FunctionSICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 5914QSKQSGQSBpiQrLubricating Gaps -examplesSwash plate axial piston machineDisplacement chamberMovable PistonRotating cylinder blockSwash plate (fixed or with adjustable angle)Valve plate (fixed)Gaps seal the displacement chamberGaps between:piston and cylinderslipper and valve platecylinder block and valve plate4SICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 5915Gap Flow Calculation5Aim:High load carrying abilityLow frictionLow leakage flowGap height lies in a range of some micrometers, whereas all otherdimensions are in a range of millimetersLaminar flow of an incompressible fluid in the gap can be described using Navier-Stokes-Equation:- neglecting mass forces- assuming steady state flow, - change of fluid velocity only in direction of gap height- pressure is not a function of gap height hAssumptions:movable surfacefixed surfacezxyh=f(x,y)hp1p2Pressure Force + Viscosity Force = 0SICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 5916Gap with constant gap heightpbhzdzppxdxx dxp1 p2xzbhPressure fieldzxpwithzvafter double integration velocity yields:21221czczxpvboundary conditions:z=0 … v=0z=h … v=0zhzxpv2216Forces applied on a fluid element:pSICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 59177Gap with constant gap heightGap flow:pbhzdzppxdxx dxp1 p2xzbh30121hbxpdzvbQhfor pressure:112pxlppplShear stress on surfaces:zvz=0z=h20hlpz2hlphzlplppxp12with:Viscous friction:lbFvPower loss due to gap flow:lphbpQPSQ23121zhzxpv221SICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 5918Gap with constant gap height8zxhp1p2p1=p2v0boundary conditions:z=0 … v=0 x=0 … p=p1z=h … v=v0 x=l … p=p2lb >> hFlow velocity:zhvv0Shear stress on surfaces:hv0Gap flow:Viscous friction:blhvblFv0Power loss due to viscous friction:200vhblvFPvSv0z 0xpzxpconstzvhvbhhvbdzzhvbdzvbQhh22020000SICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 5919xhp1p2p1v0lb >> hzp2+Gap with constant gap heightBoundary conditions:z=0 … v=0 x=0 … p=p1z=h … v=v0 x=l … p=p2Flow velocity:zhvzhzxpv0221Gap flow:bhvhbxpdzvbQh2121030Shear stress on surfaces:hvhzlppzv012221Power losses:0vFpQPPPvSvSQSlbhvhblpPS2320121922zvxpSICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 59110hrxlrGap width:dbdp1p23012hxpddzvbQhGap flow:Radial Gap with constant gapheightEquations from gap with constant gap height can be applied:10SICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 59111Gap with variable gap height11p2p1h1h2h0lv0xFpxp1 = p2 = 0112hxlhhhzxpzBoundary conditions:z=0 … v=v0x=0 … p=p1z=h … v=0 x=l … p=p2Flow velocity:00221vzhvzhzxpvp1=p2=p0= 0Gap flow:bhvhbxpdzvbQh2121030Hydrodynamic forcewithzv22zvxpSICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 59112Pressure distribution in x-direction:dxxpxpbhvhbxpQ212103from203612hvhbQxpwe obtain:dxhbQhvxp320126112hxlhhhwith:p2p1h1h2h0lv0xFpxp1 = p2 = 0pxF(1)(2)(3)(4)(5)12Gap with variable gap heightv0SICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 59113Gap with variable gap heightbaxXXnadxXnn with 111using:in our case:112 and hblhhachxlhhlhhbQhxlhhlhhvxp2112121121201212116)(Boundary conditions: x=0 p=p0hhwhere n=-2 for the first term in Eq. (4) and n= -3 for the second term in Eq. (4)13and after integration:dxhbQhvxp320126112hxlhhhwith:(4)(5)(6)(7)(8)SICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 59114Gap with variable gap height14011016phhhhbQvhhxxp21121120066hhhblQhhhlvpc0211020126phbQhvhbQhvhhlxp112hxlhhhxlhhhh112021222111016phhhhbQhhhhvhhxxpfinally we get:chxlhhlhhbQhxlhhlhhvxp2112121121201212116)(hBoundary conditions: x=0 p=p0(8)(9)(10)(11)(12)(13)(14)SICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 59115Using boundary conditions for x=l isp=p0and h=h2 from Eq. (14) follows :12120hhhhbvQ0122206phhhhhvxxpwhen h=h1=h2 p(x)=p0No load ability!011016phhhhbQvhhxxpGap with variable gap height15(14)Substituting Eq. (15) into Eq. (14) the pressure p(x) yields:(15)p2p1h1h2h0lv0xFpxp1 = p2 = 0pxFv0h1p1h2hp2xp1=p2=p0=0(16)SICFP’05, June 1-3, 2005, Linköping © Dr. Monika IvantysynovaDesign and Modeling of Fluid PowerSystems, ME 597/ABE 59116Gap with variable gap height16Load capacity due to hydrodynamicpressure field generated in the gap:021212122122ln6vhhhhhhhhlbFMaximum pressure force for h1/h2=2.2 yields:dxxhhhhhvbFdxppbFll122200006022202222max16.02.34.22.2ln2.16vhlvhlFp2p1h1h2h0lv0xFpxp1 = p2 =


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Purdue ME 59700 - Lecture notes

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