A Numerical Study on Influence of Temperature on Lubricant Film

energies

Article A Numerical Study on Inﬂuence of Temperature on Lubricant Film Characteristics of the Piston/Cylinder Interface in Axial Piston Pumps

Yueheng Song 1, Jiming Ma 2 and Shengkui Zeng 1,3,* ID

1 School of Reliability and System Engineering, Beihang University, Beijing 100191, China; [email protected] 2 Sino-French Engineering School, Beihang University, Beijing 100191, China; [email protected] 3 Science and Technology on Reliability and Environmental Engineering Laboratory, Beihang University, Beijing 100191, China * Correspondence: [email protected]; Tel.: +86-108-231-6369

Received: 13 June 2018; Accepted: 12 July 2018; Published: 13 July 2018

Abstract: The loss of kinetic energy of moving parts due to viscous friction of lubricant causes the reduction of piston pump efficiency. The viscosity of lubricant film is mainly affected by the thermal effect. In order to improve energy efficiency of piston pump, this research presents a numerical method to analyze the lubricant film characteristics in axial piston pumps, considering the thermal effect by the coupled multi-disciplinary model, which includes the fluid flow field expressed by Reynolds equation, temperature field expressed by energy equation and heat transfer equation, kinematics expressed by the motion equation. The velocity and temperature distributions of the gap flow of piston/cylinder interface in steady state are firstly numerically computed. Then the distributions are validated by the experiment. Finally, by changing the thermal boundary condition, the influence of thermal effect on the lubricant film, the eccentricity and the contact time between the piston and cylinder are analyzed. Results show that with the increase of temperature, the contact time increases in the form of a hyperbolic tangent function, which will reduce the efficiency of the axial piston pump. There is a critical temperature beyond which the contact time will increase rapidly, thus this temperature is the considered as a key point for the temperature design.

Keywords: thermodynamic; numerical simulation; thermal effect; axial piston pumps

1. Introduction As sliding pairs of axial piston pump move, both the metal parts and lubricants heat up, which causes temperature to increase significantly, decreases oil viscosity, reduces the bearing capacity of the oil, and intensifies the radial movement of the parts simultaneously [1]. The shape of the oil film changes, as does the lubrication mode, and the trajectory of piston could vary with the changing of oil film. The temperature impacts on oil film characteristics are important for analysis of the efficiency of the sliding pair. Practical experience also indicates that the medium temperature significantly affects life. Cai et al. [2] tested the life of steel materials with a lubricating medium in the range of 23–175 ◦C and verified that the wear degree was directly related to the temperature. Specifically, the life of steel materials was closely related to the oil temperature distribution in the gap, and the distribution condition had a distinct influence on the oil characteristics. Unfortunately, the general test method could take temperature as the only variables; thus the results obtained by this way could only indicate the existence of a fuzzy relationship between temperature and life.

Energies 2018, 11, 1842; doi:10.3390/en11071842 www.mdpi.com/journal/energies Energies 2018, 11, x FOR PEER REVIEW 2 of 16 Energies 2018, 11, 1842 2 of 16

To indicate the relationship between the temperature and life more precisely, the behavior of lubricantTo indicate film between the relationship the moving between pair need thes temperatureto be studied. and Because life more the moving precisely, process the behavior is affected of lubricantby fluid, filmsolid, between and thermal the moving conditions pair needs together, to be it studied. is a typical Because multiple the moving-domain process coupled is affected problem. by fluid,Many solid, researchers and thermal have conditionsstudied this together, problem it is around a typical the multiple-domain bearing. McCallion coupled et al. problem. [3] solved Many the researchersReynolds and have e studiednergy equations this problem separately around the and bearing. neglected McCallion the effects et al. [ 3] of solved pressures the Reynolds during and the energycalculation equations of temperature separately distribution and neglected. Ferron the effects et al. of[4] pressures thoroughly during studied the calculationthe thermohydrodynamic of temperature distribution.(THD) performance Ferron etof al.a plain [4] thoroughly journal bearing studied and the determined thermohydrodynamic its thermal (THD)characteristics. performance Rohde of et a plainal. [5] journal studied bearing the elastic and determinedand thermal its deformations thermal characteristics. in a slider Rohdebearing, et al.noted [5] studiedthat the the variations elastic and in thermalfluid viscosity deformations with temperature in a slider bearing, were much noted more that influential the variations than in solid fluid deformation viscosity with due temperature to thermal wereand elastic much moreeffects. influential Researchers than ar solide still deformation interested in due the to journal thermal bearing and elastic where effects. oil or Researchers gas is used are as stilllubricants interested [6,7]. in Lu the et journal al. [8] conducted bearing where a thermal oil or- gasfluid is coupling used as lubricants study on characteristics [6,7]. Lu et al. [of8] air conducted–oil two aphase thermal-fluid flow and coupling heat transfer study onin acharacteristics micro unmanned of air–oil aerial two vehicle phase flow (UAV and) bearing heat transfer chamber. in a micro Some unmannedresearchers aerial took vehiclethe cavitation (UAV) bearinginto account chamber. [9] and Some others researchers develop tooked an the efficient cavitation numerical into account method [9] and[10].others developed an efficient numerical method [10]. Axial piston piston pumps pumps with with constant constant pressure pressure are arethe main the main type typeof engine of engine-driven-driven pumps pumps(EDP), (EDP),whose whose lifetime lifetime is affected is affected by three by three sliding sliding pairs, pairs, specifically specifically the the swash swash plate/slipper pair, cylinder/valvecylinder/valve plate plate pair, pair, and piston/cylinder pair, pair, as as shown shown in in Figure Figure 11..

Figure 1.1. Sliding pairs in axial piston pump.pump.

Several Elasto Hydrodynamic Lubrication ( (EHL)EHL) researches have focused on the piston pump. Gels etet al.al. developed a simulation tool based on thethe ReynoldsReynolds equation,equation, which allowed to vary thethe geometry of the slidingsliding parts. With the help of thisthis tool,tool, GelsGels havehave foundfound an optimaloptimal compromise of piston/cylinderpiston/cylinder interface which reduces the losses ofof energy [[11].11]. MaMa etet al. presented a method onon thethe basis ofof thethe EHL EHL model model to to analyze analyze the the behavior behavior of of the the swash swash plate/slipper plate/slipper pair pair [12 [12].]. Olems Olems presented presented an analyticalan analytical model model that integrates that integrates the multiphysics the multiphysics characteristic characteristic equations equations including including Reynolds Reynolds Equation andEquati energyon and equation energy [ 13equation]. Given [13]. inlet Given medium inlet temperature medium temperature and operating and conditions, operating the conditions, temperature the distributiontemperature in distribution the clearance in of the piston/cylinder clearance of piston/cylinder interface wasdetermined interface was using determined the model. using A series the ofmodel. studies A of series MAHA of studies research of center MAHA were research carried center out based were on carried Olem’s out results. based Ivantysynova on Olem’s results. et al. establishedIvatysynova a et complete al. established model ofa complete main sliding model pairs of main for thermal sliding analysis pairs for of thermal oil film analysis gap [14]. of Kazama oil film establishedgap [14]. Kazama a non-isothermal established model a non- ofisothermal slipper/swash model plate of slipper/swash pair based on plate thermos-hydraulic pair based on thermos dynamic- theoryhydraulic [15 ].dynamic Pelosi, Ivantysynova theory [15]. Pe [16losi,], and Ivatysynova Zecchi [17 ][16], built and an EHLZecchi model [17] built for the an piston/cylinder EHL model for pair the andpiston/cylinder the swash plate/slipper pair and the pair.swash However, plate/slipper as shown pair. in However, all these studies,as shown few in studies all these have studies, modelled few thestudies interaction have modelled relationship the among interaction elastic, relationship thermal and among hydraulic elastic, behaviors thermal only and use hydraulic numerical be methodhaviors withoutonly use establishing numerical method the computer-aided without establishing design (CAD) the computer model or-aided using design commercial (CAD software.) model or using commercialThis study software. was designed to investigate the temperature distribution for the further accurate analysisThis on study the thermal was designed effect on to the investigate piston/cylinder the temperature interface. distributionThe flow velocity for the distribution further accurate in the piston/cylinderanalysis on the thermal interface effect at a steady on the operating piston/cylinder state of interface. the piston The pump flow (the velocity output distribution pressure of pumpin the ispiston/cylinder steady) was calculated interface at using a steady the Reynolds operating equation. state of the Then, piston the pump velocity (the distribution output pressure was imported of pump intois steady) the model, was calculated considering using the the comprehensive Reynolds equation. effects ofThen flow, the and velocity temperature distribution distributions was imported in the gap,into the and model the movement, considering of moving the comprehensive components. effects of flow and temperature distributions in the gap, and the movement of moving components.

Energies 2018, 11, 1842x FOR PEER REVIEW 3 of 16

Only the numerical method is used during the whole simulation process. So, a parametric simulationOnly the model numerical without method the construction is used during of thea three whole-dimensional simulation process. (3D) digital So, a model parametric was established,simulation model which without is convenient the construction for further of a application three-dimensional to the (3D) simulation digital model of sliding was established, pairs with differentwhich is convenientstructures. for further application to the simulation of sliding pairs with different structures.

2. Description Description of Model Under the assumption that the gap clearance (radial direction) was much smaller than the other two dimensions dimensions (circumferential (circumferential and axial directions), directions), an unwrapped unwrapped Cartesian Cartesian reference reference system system was was used for the description of the computation domain. The The coordinates coordinates in in this this reference system are defineddeﬁned as Equation (1) and presented in Figure2 2..   xx =θθ   yyr= r (1(1))  z = z zz

Figure 2. Definitions of coordinates and gap height. Figure 2. Deﬁnitions of coordinates and gap height.

2.1. Mathematical Model 2.1. Mathematical Model The axial piston pump has an odd number of pistons arranged in a circular array within a The axial piston pump has an odd number of pistons arranged in a circular array within a housing housing that is commonly referred to as a cylinder block. This cylinder block is driven to rotate about that is commonly referred to as a cylinder block. This cylinder block is driven to rotate about its axis of its axis of symmetry by an integral shaft that is aligned with the pumping pistons. A piston/cylinder symmetry by an integral shaft that is aligned with the pumping pistons. A piston/cylinder interface interface is an assembly with one piston, one cylinder, and a film of lubricant oil. When the piston is an assembly with one piston, one cylinder, and a film of lubricant oil. When the piston pump is pump is operating, the piston moves in a linear reciprocating pattern within the cylinder block, operating, the piston moves in a linear reciprocating pattern within the cylinder block, suctioning or suctioning or discharging oil. discharging oil. Analysis of thermodynamic behavior of piston/cylinder interface in piston pumps is a fluid- Analysis of thermodynamic behavior of piston/cylinder interface in piston pumps is a structure interaction problem. The motion for fluid flow, heat transfer, and the motion of the piston fluid-structure interaction problem. The motion for fluid flow, heat transfer, and the motion of are expected to be determined simultaneously. Because it is a time-dependent problem, one shaft the piston are expected to be determined simultaneously. Because it is a time-dependent problem, revolution is divided into a number of discrete time steps. For each time step, the problem was one shaft revolution is divided into a number of discrete time steps. For each time step, the problem considered time-independent. was considered time-independent. The momentum transport for the fluid flow in the gap is described by the Reynolds equation The momentum transport for the fluid flow in the gap is described by the Reynolds equation [18]: [18]: 3 3 ∂ ∂phpph33 ∂ ∂ p h h    h∂h  h∂h  h ∂h + = 662(UUVVUh − U0) + (V h − V0)  + 2 (2) x x z z   hh00 x   z t (2) ∂ ∂xxµ μ ∂ zz∂  µμ   x∂  z∂  t ∂ where h is the filmfilm thickness, thickness, μµ is the oil oil viscosity, viscosity, UUhh is fluidfluid velocity component along the circumferential direction on the boundary y = h, U0 isis fluid fluid velocity component along along the circumferential direction on the boundary y = 0, Vh isis fluid fluid velocity component along along the the axial direction on the boundary y = h,, V00 isis fluid fluid velocity velocity component component along along the the axial axial direction direction on on boundary boundary y = 00.. yy = hh indicaindicatestes the the interface interface between between the the cylinder cylinder and and oil, oil, and and yy = 0 indicates the interface between the piston and oil. We assumed that the velocity of oil at oil/solid interface was the same as the solid. So Uh, U0, Vh, and V0 were constant.

Energies 2018, 11, 1842 4 of 16 between the piston and oil. We assumed that the velocity of oil at oil/solid interface was the same as the solid. So Uh, U0, Vh, and V0 were constant. The temperature ﬁeld of the ﬂuid is governed by the convection-conduction equation [18]:

∂(ρ T ) c o o + ρ c div(V · T ) = k div(gradT ) + φ (3) o ∂t o o o o o o where, To is oil temperature, co is the specific heat capacity at constant pressure of oil, ρo is the density of oil, ko is the thermal conductivity of oil, and φo is the viscous dissipation term. The vector V in Equation (3) is the velocity of fluid composed by u and v which are determined by followed equations: 1 ∂p 2 y u = 2µ ∂x y − yh + (Uh − U0) h + U0 (4) 1 ∂p 2 y v = 2µ ∂z y − yh + (Vh − V0) h + V0 The temperature depends on the fluid velocity, so the last term in Equation (3) describes the energy dissipation: " # ∂u 2 ∂v 2 φ = µ + (5) o ∂y ∂y

Then the local viscosity of grid point in the gap can be calculated using the temperature distribution with the Reynolds viscosity temperature equation [18]:

−σ(To−Tref) µ = µrefe (6) where Tref is the reference temperature, µref is the viscosity of oil at reference temperature, and σ is the viscosity-temperature index. The local viscosity is then used as new value in the following iteration cycle of algorithm. For the temperature distribution inside the piston, the temperature distribution was determined using the heat conduction equation, which is written as Equation (7) [18]: ∂ ρpTp c = k divgradT + φ (7) p ∂t p p p where Tp is the piston temperature, cp is the speciﬁc heat capacity at constant pressure of piston, ρp is the density of piston, kp is the thermal conductivity of piston, and φp is the viscous dissipation term. Because the cylinder is geometrically much larger than the gap and the piston, the temperature is here considered a constant, Tc (case temperature).

2.2. Boundary Conditions To simplify the model, only one piston/cylinder interface was considered. This piston/cylinder formed the system boundary of the thermal calculation, as shown in Figure3. Because the nine piston/cylinder interfaces were symmetrical around the distribution circle, the behavior of these interfaces could be expected to be almost the same. For this reason, it was here considered reasonable to choose one interface as a representative. The numbers here represent the corresponding boundaries as labeled in Figure3. Energies 2018, 11, 1842 5 of 16 Energies 2018, 11, x FOR PEER REVIEW 5 of 16

FigureFigure 3. 3.Computation Computation domain domain and and boundary. boundary.

TheThe boundary boundary conditions conditions of of fluid fluid mechanics mechanics equations equations (Reynolds (Reynolds equation) equation) are are shown shown in in EquationEquation (8): (8): ( ◦ ◦  Psuc(θs =0− 180 ) Interface 3 : P| =  Psuc θs 0 180  Interfacebottom 3: P edge  P (θ = ◦ − ◦) bottom edge dis s 180 360 (8) Pdis θs 180 360  (8) InterfaceInterface 4 : P|top 4: edgePP= Pcase top edge case ◦ ◦ where Psuc is the output pressure of the pump when the piston is in the oil suction area (θs = 0 –180 ), where Psuc is the output pressure of the pump when the piston is in the oil suction area (θs = ◦0°–180°),◦ Pdis is the output pressure of the pump when the piston is in the oil discharge area (θs = 180 –360 ), Pdis is the output pressure of the pump when the piston is in the oil discharge area (θs = 180°–360°), and Pcase is the case pressure of the pump. and Pcase is the case pressure of the pump. The boundary conditions of thermal analysis are the environmental temperature and temperatures The boundary conditions of thermal analysis are the environmental temperature and of the medium at the inlet and outlet temperatures. For one piston/cylinder interface, solving the temperatures of the medium at the inlet and outlet temperatures. For one piston/cylinder interface, energy equation of fluid and heat conduction equation of solid covers the changes in heat in one phase solving the energy equation of fluid and heat conduction equation of solid covers the changes in heat and between two phases. For each specific boundary, the following boundary conditions are defined. in one phase and between two phases. For each specific boundary, the following boundary conditions The numbers here represent the corresponding boundaries as labeled in Figure3. are defined. The numbers here represent the corresponding boundaries as labeled in Figure 3. (1)(1) OnOn the the film-cylinder film-cylinder interface: interface: ∂ToT(x, xz,,) z  k kc  TT xo(x z,z) T− Ta oa = − c hh x(,x z,z) o (9) ∂y dc kodp ∆y (9) yy= d2 k d y y c op  2

(2) On the ﬁlm-piston interface: (2) On the film-piston interface:

∂ToT(x, z x), z kkp ∂ TTp( xx, z, z) o   = − pph(x, z)   (10) dp  hx ,z ∂y y= kodp ∂y piston−ﬁlm (10) y2dp kop d y y piston-film 2

(3) Oil on the bottom edge of the ﬁlm: (3) Oil on the bottom edge of the film:

TTTo|bottom edge = Tmix (11) obottom edge mix (11)

wherewhereT Tmixmix isis thethe mixingmixing temperature,temperature, whichwhich isis determineddetermined basedbased onon thethe energy energy balance balance [ 19[19].]. (4)(4) OilOil on on the the top top edge edge of of the the ﬁlm: film: To|top edge = Tin (12) TT otop edge in (12) where Tin is inlet temperature of oil. where Tin is inlet temperature of oil. (5) On the bottom surfaces of the piston:

Energies 2018, 11, x FOR PEER REVIEW 6 of 16 Energies 2018, 11, 1842 6 of 16

Tpp x, y ηd  T x, y  T (13)  pz0 mix  (5) On the bottom surfaces of the piston:zk2 p z0 ∂T (x, y) ηd where η is piston convection heatp transfer = coefficient.− p ( ) − Tp x, y z=0 Tmix (13) ∂z 2kp (6) On the top of the piston: z=0

where η is piston convection heat transfer coefﬁcient. Tpp x, y ηd   Tp x, y  T in (14) (6) On the top of the piston: zl zk2 p t zl t

∂Tp(x, y) ηdp = − Tp(x, y) − Tin (14) Once the oil film temperature distribution∂z was2 determined,k mechanicalz=lt analysis could progress. z=lt p 2.3. Deformation Equation Once the oil ﬁlm temperature distribution was determined, mechanical analysis could progress. For an accurate description of thermodynamic behavior of piston/cylinder interface, it is not sufficient2.3. Deformation to just solve Equation the energy and Reynolds equation, but one also needs to consider deformation due toFor temperature an accurate and description pressure. In of this thermodynamic paper, the deformation behavior ofmatrix piston/cylinder method was interface, used to calculate it is not thesufﬁcient elastic to deformation just solve the and energy thermal and Reynolds deformation equation, of metal but parts. one also The needs deformation to consider equations deformation are showndue to in temperature Equation (15) and [20] pressure.: In this paper, the deformation matrix method was used to calculate the elastic deformation and thermal deformation of metal parts.ij, The deformation equations are shown h i,[] j  p  p C  A E, f g cfg, p fg, in Equation (15) [20]: fg i,j (15) h (i, j) = [p + p | ] C ij, ∆A E h i,[] j∑∑ f ,g Tc Tf ,g Cp f ,g T  f g PTf,, g ref f g fg (15) = [ − ] i,j hT(i, j) ∑ ∑ TP| f ,g Tref CT| f ,g g where hE is the elastic deformation, pf,g is thef oil pressure at node (f, g), pc|f,g is the contact pressure, ij, h p f g p whichwhere is Ezerois the in elasticthis case deformation,, because the pistonf,g is the bears oil pressureno force in at the node initial ( , ), freec| casef,g is, theCp contact is the pressure, elastic fg ,i,j which is zero in this case, because the piston bears no force in the initial free case, C is the elastic deformation matrix, which represents the displacement of node (i, j) caused by the punit f ,g load acting ondeformation the node (f matrix,, g). which represents the displacement of node (i, j) caused by the unit load acting on the node (f, g). ij, Similarly, hT is the thermal deformation, C i,j is the thermal deformation matrix, TP|f,g is the T fg, Similarly, hT is the thermal deformation, CT| f ,g is the thermal deformation matrix, TP|f,g is the ◦ temperaturetemperature at at node node ( (ff,, gg),), and and TTrefref isis the the reference reference temperature temperature and and set set to to 25 25 °CC..

2.4.2.4. Force Force A Analysisnalysis TheThe force mustmust be be analyzed analyzed to solveto solve motion motion equations. equations. To clarify To clarify the description the description of force of analysis, force analysis,some deﬁnitions some definitions are presented are presented in Figure in4. Figure 4.

FigureFigure 4. 4. DefinitionsDeﬁnitions used used for for mechanical mechanical analysis. analysis.

WhenWhen the the pump pump is is operating, operating, the the cylinder cylinder block block slides slides over over the the valve valve plate. plate. With With the the change change in in position within the distribution circle (θs), the displacement chambers of each piston are connected position within the distribution circle (θs), the displacement chambers of each piston are connected to the low-pressure port (θs = 0°◦–180°)◦ or the high-pressure port (θs = 180°◦–360°).◦ The pistons were to the low-pressure port (θs = 0 –180 ) or the high-pressure port (θs = 180 –360 ). The pistons were

Energies 2018, 11, x FOR PEER REVIEW 7 of 16 Energies 2018, 11, 1842 7 of 16 joined to the slippers that glided over the swash plate. By changing the swash plate angle β, the piston stroke s could be adjusted. θs, lt, and s were subject to Equation (16). joined to the slippers that glided over the swash plate. By changing the swash plate angle β, the piston  stroke s could be adjusted. θs, lt, and s were subjectθs toωt Equation (16). d l s 1 cosθ  tan β ts2 θs = ωt (16) l = ds (1 − cos θ ) tan β (16) t 2 ds s skl00  lts  l ds 1  cosθ  tan β  s = l0 − lt = l0 − 22 (1 − cos θs) tan β · k

TheThe loadsloads actedacted onon thethe pistonpiston inin thethe cavitycavity ofof piston/cylinderpiston/cylinder interface in the specific speciﬁc coordinates systemsystem isis shownshown inin FigureFigure5 5. .

FigureFigure 5.5. Graph of mechanical analysis.analysis.

BecauseBecause of the overturning overturning angle angle αα, ,the the velocity velocity of of piston piston ṡ shad˙ had a component a component normal normal to tothe the ω, c ωso, the so theCoriolis Coriolis force force F tookFc took place. place. With With the movementthe movement of cylinder of cylinder block block around around the shaft,the shaft, the u thecentrifugal centrifugal force force F Factedu acted on onthe the center center of ofgravity gravity of of the the piston. piston. Because Because the the piston piston movement movement took place within six degrees of freedom, there was also the inertia force in z-direction Fi and the inertia place within six degrees of freedom, there was also the inertia force in z-direction Fi and the inertia moment Mi. All these forces acted on the center of gravity and were defined using Equation (17): moment Mi. All these forces acted on the center of gravity and were deﬁned using Equation (17):

Fc  2mω  s . Fc = −2mω × s 22 ds 0 FFu = mω j j' u 2 (17) ( . (17) FFsi =−mms  i . Mi = −Jω Mi Jω

TheThe swashswash plateplate waswas removedremoved forfor thethe calculationcalculation ofof actingacting forceforce ofof slipperslipperF Fss.. FFss waswas resolvedresolved throughthrough thethe force force equilibrium equilibrium equation equation in in the thex-direction. x-direction.Fs wasFs was decomposed decomposed into into the ythe-direction y-direction and zand-direction z-direction as presented as presented in Equation in Equation (18): (18):

FkF cos α β  F sz= F s cos(α + β) · k sz s (18(18)) FsyFjsy=FFsssin sin(αα+ β) · j

TheThe pressure pressure force force FFpp waswas calculated calculated using using the the pressurepressure of of displacement displacement chamber chamber PPc,c, whichwhich ◦ ◦ ◦ ◦ differeddiffered betweenbetween thethe low-pressurelow-pressure portport ((θθss= = 00°–180–180°)) and the high-pressurehigh-pressure portport ((θθss == 180180°––360360°))::

2 πd 2 πdp (19) FFkp= Pc · k (19) p 4 c TheThe actingacting forceforce ofof oiloil waswas normalnormal toto thethe pistonpiston surfacesurface andand isis deﬁneddefined asas follows:follows:

2π l d Z 2π Z tl p Fjtp x, z   cos xd dzdxp (20) Fo =o 00p(x, z) · j · cos x2 dzdx (20) 0 0 2

Due to the movement of piston, the friction of oil Ff acted on the piston. Ff was calculated using Equation (21):

Energies 2018, 11, 1842 8 of 16

Due to the movement of piston, the friction of oil Ff acted on the piston. Ff was calculated using Equation (21):

! . ! ! Z 2π Z lt ∂p(x, z) ∂p(x, z) µs dp Ff = −h(x, z) + + dzdx · k (21) 0 0 dp ∂z h(x, z) 2 2 ∂x

Because the forces deﬁned by Equations (17) and (19) acted through the center of gravity of the piston, their resultant moments were all zero. This left the moments produced by Fs, Fo, and Ff, which can overturn the piston. These moments were calculated using Equations (22)–(24):

MFs = Fs · ls sin(α + β) (22)

Z 2π Z lt dp MFo = p(x, z) · j· cos x(z − lm) dzdx (23) 0 0 2 ! . ! 2 ! Z 2π Z lt ∂p(x, z) ∂p(x, z) µs d = − + + p · MFf h(x, z) cos x dzdx k (24) 0 0 dp ∂z h(x, z) 2 2 ∂x where p(x,z) in Equations (20), (21), (23) and (24) is the pressure distribution obtained by Section 2.1. h(x,z) will be determined in Section 2.6. Assuming the vector sum of forces and moments are Fsum and Msum.

F = F + F + F + F + F + F + F sum c u i s p o f (25) = + + + Msum Mi MFs MFo MFf

2.5. Movement Analysis The movement of the piston was analyzed during force analysis. According to Newton’s second law, several types of force were identiﬁed. They are shown in Equation (26).

Fsum ve|t+dt = ve|t + m dt

α| t+dt = α| t + vα|tdt

2.6. Calculation of the Thickness of the Oil Film The thickness of the oil film h(x,z) was determined as described below. As shown in Figure6, supposing that the space between the piston and cavity was filled with an oil film, the thickness of that oil film distribution could be determined according to the geometry 0 characteristics of piston and cavity. In section C-C, O1 was the center of the cavity, O1 was the projection of O1 on the y axis and O2 was the center of piston, assuming that the piston overturning angle was α and projection of eccentric distance at bottom of piston e on y axis was e0. Piston shaft and cavity shaft intersected in the position (0, 0, l1)|(x,y,z). The following geometric relationships were established: e0 = l1 tan α (27)

For an arbitrary cross section C-C of piston, O1 and O2 were used to deﬁne a straight line that crossed the surface of the piston at P1 and crossed the cavity surface at P2. Any position on surface of piston P1 could correspond to the circumferential angle θ. |P1P2| was the thickness of the oil ﬁlm, whose value is here written as h(θ, z). In order to simplify the calculation, there was a transformation Energies 2018, 11, 1842 9 of 16

of system of coordinates. As shown in Figure6b, O 2 was located at (0, 0, z) and O1 at (-e2,-e1, z). e1 and e2 were subject to Equation (28):

0 e1 = |O1 O2| = (z − l1) tan α e1 (28) e2 = tan θ

In this way, the algebraic expression of straight line determined by O1O2 was as follows:

y = x tan θ (29)

On the cross section C-C, cavity was an ellipse, and the piston was a circle. Their algebraic expression was as shown in Equation (30):

2 2 4(x+e2) + 4(y+e1) = d 2 d 2 sec2 α 1 c c (30) 2 2 2 dp x + y = 4

The coordinates of P1 and P2 were determined by solving Equations (29) and (30) simultaneously. The results are presented in Equations (31) and (32):  dp  xP = cos θ 1 2 (31)  = dp yP1 2 sin θ

 r 2 2  x = dc sec α − e  P2 4(sec2 α+tan2 θ) 2 (32) r 2 2  y = dc sec α tan θ − e  P2 4(sec2 α+tan2 θ) 1

Oil film thickness h(θ, z) under specific overturning angle α was determined using the coordinates of P and P : 1 2 q = − 2 + − 2 h(x, z) xP1 xP2 yP1 yP2 (33) Along with the elastic and thermal deformations, oil film thickness should be written as:

h(x, z) = h(x, z) + hE(x, z) + hT(x, z) (34) Energies 2018, 11, x FOR PEER REVIEW 9 of 16

FigureFigure 6. 6.Graph Graph ofof calculationcalculation of of oil oil ﬁlm film thickness: thickness: ( a(a)) Axial Axial section section view; view; ( b(b)) Bottom Bottom view. view.

3. SimulationFor an arbitrary Algorithm cross and section Conditions C-C of piston, O1 and O2 were used to define a straight line that crossed the surface of the piston at P1 and crossed the cavity surface at P2. Any position on surface of Simulation of the thermodynamic behavior of the piston requires knowledge of the eccentric piston P1 could correspond to the circumferential angle θ. |P1P2| was the thickness of the oil film, position, velocity of the piston, and all forces acting on it. Among these, eccentric position and forces whose value is here written as h θ,z . In order to simplify the calculation, there was a transformation

of system of coordinates. As shown in Figure 6b, O2 was located at (0, 0, z) and O1 at (-e2, -e1, z). e1 and e2 were subject to Equation (28):

e1O 1 'O 2  z  l 1  tanα e (28) e  1 2 tanθ

In this way, the algebraic expression of straight line determined by O1O2 was as follows:

yx tanθ (29)

On the cross section C-C, cavity was an ellipse, and the piston was a circle. Their algebraic expression was as shown in Equation (30):

22 44x e  y e  211 dd2 2sec 2 α cc (30) d 2 xy22p 4

The coordinates of P1 and P2 were determined by solving Equations (29) and (30) simultaneously. The results are presented in Equations (31) and (32):

 dp xP  cosθ  1 2  (31)  dp yP  sinθ  1 2

 d 22sec α  c xeP2  2 22  4 secα  tan θ  (32) 22  dc sec α yP1tanθ e 2 4 sec22α  tan θ   

Oil film thickness h θ,z under specific overturning angle α was determined using the

coordinates of P1 and P2:

22     (33) h x, z  xPPPP x  y y  1 2 1 2

Energies 2018, 11, x FOR PEER REVIEW 10 of 16

Along with the elastic and thermal deformations, oil film thickness should be written as:

hxz ,,,,  hxz   hxzET   hxz  (34)

3. Simulation Algorithm and Conditions Energies 2018, 11, 1842 10 of 16 Simulation of the thermodynamic behavior of the piston requires knowledge of the eccentric position, velocity of the piston, and all forces acting on it. Among these, eccentric position and forces are profoundly inﬂuencedinfluenced by temperaturetemperature distribution withinwithin the gap. An iterative method for the simultaneous calculation of all the requirements was here developed. Figure 77 showsshows aa diagramdiagram ofof these calculations.

Figure 7. Flow chart of calculation. Figure 7. Flow chart of calculation.

With constant inlet and outlet pressure, the velocity distribution is considered constant and a basic condition of the Reynolds equation. When the velocity results were considered as the input of energy equation, the temperature distribution indicated by resolving of energy equation was the temperature distribution of oil film in steady state. The mechanical analysis was completed considering hydrodynamic forces and temperature distribution as inputs of motion equations. Using the overrelaxation iterative formula of the finite difference method, the Reynolds equation and the Energies 2018, 11, 1842 11 of 16 energy equation are solved by programming in MATLAB. If the absolute error of the results on all interior points obtained by two adjacent iterations is not greater than that of the prescribed error, then the iterative solution is considered to be convergent. An axial piston pump with nine pistons was here selected for study. To start the simulation, some fundamental conditions were defined, which were the structure and working parameters of the pump, the material properties and thermal coefficients of the piston/cylinder interface, and the pressure of displacement chamber. The structure and working parameters of a specific type of pump are presented in Table1.

Table 1. Structure and working parameters.

Parameter Value Number of pistons 9 Rated working pressure 225 bar Rated rotation speed 1000 rpm Distribution circle diameter 80 mm Piston diameter 30.14 mm Cylinder cavity diameter 31.2 mm

The material properties and thermal coefﬁcients of the pump are shown in Table2[21].

Table 2. Material properties.

Heat Convection Density Young Modulus Speciﬁc Heat Heat Conductivity Viscosity Coefﬁcient (kg/m3) (Gpa) Capacity (J/(kg·K)) (W/(m·K)) (20◦C) (Pa·s) (W/(m2·K)) Cylinder 8100 130 390 41.9 / / Piston 7850 210 490 50.28 / 50 Oil 862.1 / 1880 0.143 0.15 /

4. Results and Discussion A simulation of the chosen piston pump was conducted with the model and numerical algorithm introduced in this paper. In order to validate the developed model in this article, the simulated velocity and temperature distribution were compared with the experiment performed by Pelosi [22]. Constant inlet and outlet ◦ ◦ pressures on both sides of the piston when the pump was in normal operation were 16 bar (θs = 0 –180 ) ◦ ◦ and 225 bar (θs = 180 –360 ) (boundary z = 0 in Figure8) and 5 bar (boundary z = 1 in Figure8) respectively. For the temperature simulation, the input parameters were same as Pelosi [22], which are 57.5 ◦C for the case temperature, 48 ◦C for temperature at high pressure port, and 43 ◦C for temperature at low pressure port. ◦ We selected the state where lt = l0, where the piston was in the position of θs = 0 on the distribution circle, and the whole piston was in the cylinder, The corresponding α = 0.0067◦. Figure8 shows the velocity distribution of oil film at the interface of the oil and the piston. Figure9a shows the result of thermal simulation, and Figure9b shows the measurement result of the oil film at the interface of the oil and the piston carried by Pelosi [22]. Thus, we can conclude that the model could describe the oil temperature distribution, with reference to previous research. Results indicated that the velocity distribution of oil film in piston/cylinder interface was affected not only by the relative velocity between piston and cylinder in the form of sinusoidal wave (z direction) but also the rotation of piston (x direction). The motion of the piston would lead the oil to follow its motion, and the change of the velocity direction of the oil would have hysteresis because of the viscous effect, so the vortex would be produced. Energies 2018, 11, x FOR PEER REVIEW 12 of 16

Energies 2018, 11, 1842 12 of 16 Energies 2018, 11, x FOR PEER REVIEW 12 of 16

Figure 8. Velocity distribution.

Results indicated that the velocity distribution of oil film in piston/cylinder interface was affected not only by the relative velocity between piston and cylinder in the form of sinusoidal wave (z direction) but also the rotation of piston (x direction). The motion of the piston would lead the oil to follow its motion, and the change of the velocity direction of the oil would have hysteresis because of the viscous effect, so the vortex would be produced. Analysis of the Figure 8 also showFigureed that8. Velocity the velocity distribution of flow. near the boundaries (z = 0 and z = Figure 8. Velocity distribution. 1) was faster than in the middle. Results indicated that the velocity distribution of oil film in piston/cylinder interface was affected not only by the relative velocity between piston and cylinder in the form of sinusoidal wave (z direction) but also the rotation of piston (x direction). The motion of the piston would lead the oil to follow its motion, and the change of the velocity direction of the oil would have hysteresis because of the viscous effect, so the vortex would be produced. Analysis of the Figure 8 also showed that the velocity of flow near the boundaries (z = 0 and z = 1) was faster than in the middle.

FigureFigure 9. 9. ((aa)) Simulated temperature distribution; ( (b)) Measured temperature temperature distribution distribution by by experimentexperiment [[2222].].

AnalysisComprehensive of the Figure analysis8 also of showedthe contents that theof Figure velocitys 8 ofand flow 9 indicate near thed boundariesthat heat transfer ( z = 0 wa ands fasterz = 1) waswhere faster the thanvelocity in the wa middle.s faster. Temperature diffused along the velocity, whether the temperature was highComprehensive or low. This indicate analysisd ofthat the thecontents accurate of Figures calculation8 and9 ofindicated the velocity that heat distribution transfer was wa fasters very whereimportant the velocity to the wascalculation faster. Temperature of temperature diffused distrib alongution, the considering velocity, whether the velocity the temperature has a highly was highsignificant orFigure low. effect This9. (a indicated)on Simulated temperature that temperature the diffusion. accurate distribution; calculation of(b)the Measured velocity distributiontemperature was distribution very important by to theexperimentThe calculation higher [2 temperature2 of]. temperature region distribution, correspond consideringed to a region the where velocity the has fluid a highlyfilm wa significants thinner, where effect onmore temperature heat was diffusion.generated. In these areas, the oil film tended to become thinner in next time step becauseTheComprehensive high higher temperature temperature analysis reduced region of the the correspondedcontents carrying of Figurecapacity to as region 8 ofand oil where9 under indicate the thed fluid condition thatfilm heat was transferof lower thinner, wa viscosity.s where faster morewhereThe eccentricity heat the velocity was generated. may wa s be faster. increased, In Temperature these areas,but it thestilldiffuse oil depend filmd along tendeded theon velocity, the to become structure whether thinner and the the in temperature pressure next time in stepwa thes becausehighchamber. or high low. temperature This indicate reducedd that the the carrying accurate capacity calculation of oil of under the velocitythe condition distribution of lower wa viscosity.s very important to the calculation of temperature distribution, considering the velocity has a highly The eccentricity may be increased, but it still depended on the structure and the pressure in the chamber. significant effect on temperature diffusion. 5. AnalysisThe higher of Thermal temperature Effect region corresponded to a region where the fluid film was thinner, where more heat was generated. In these areas, the oil film tended to become thinner in next time step With the thermal results, it was possible to calculate the fluid pressure and piston motion in the because high temperature reduced the carrying capacity of oil under the condition of lower viscosity. piston/cylinder interface. In order to investigate the effect of inlet oil temperature on the lubricating The eccentricity may be increased, but it still depended on the structure and the pressure in the chamber.

Energies 2018, 11, x FOR PEER REVIEW 13 of 16 Energies 2018, 11, 1842 13 of 16 5. Analysis of Thermal Effect characteristicsWith the thermal of oil ﬁlm, results, we usedit was the possible model to proposed calculate inthe this fluid paper pressure to simulate and piston the movementmotion in the of thepiston/cylinder bottom of piston interface. in the In order chamber to investigate at different the inlet effect temperatures. of inlet oil temperature To facilitate on the the simulation,lubricating wecharacteristics assumed that of oil the film, temperatures we used the at m highodel pressure proposed port in this and paper low pressureto simulate port the were movement the same of the as ◦ ◦ ◦ ◦ ◦ thebottom inlet of temperature piston in theTin , chamber and the T atin differentof the six inlet simulations temperatures. was 20 ToC, facilitate 40 C, 60 theC, simulation, 80 C, 100 weC, ◦ ◦ andassume 140d Cthat respectively, the temperatures and the at case, high temperature pressure portTc andwas low always pressure 50 C. port were the same as the inlet temperatureFigure 10 T presentsin, and the simulated Tin of the trajectory six simulations of the piston was 20 bottom °C, 40 in °C, the 60 cavity °C, 80 over °C, one 100 shaft °C , and revolution 140 °C withrespectively, different and inlet the temperatures. case, temperature The following Tc was always conclusions 50 °C.can be summarized:

FigureFigure 10. 10. Simulated trajectory of the piston bottom in the the cavity cavity over over one one shaft shaft revolution revolution with with differentdifferent inletinlet temperatures.temperatures.

Energies 2018, 11, x FOR PEER REVIEW 14 of 16

Energies 2018Figure, 11, 184210 presents simulated trajectory of the piston bottom in the cavity over one14 of shaft 16 revolution with different inlet temperatures. The following conclusions can be summarized: (1). When the piston rotates within the suction zone (θs =◦ 0°–180°),◦ the oil pressure of the cylinder (1). When the piston rotates within the suction zone (θs = 0 –180 ), the oil pressure of the cylinder cavitycavity is is smaller, smaller, therefore therefore the the oil oil film film supporting supporting force force is is smaller, smaller, and and the the radial radial movement movement of of the the pistonpiston intensifies intensifies under under the the same same inertial inertial force force compared compared with with the the condition condition when when the the pistons pistons rotates rotates into the outlet zone (θs = 180°◦ –360°).◦ The reduction of the eccentricity during the discharge area is into the outlet zone (θs = 180 –360 ). The reduction of the eccentricity during the discharge area is causedcaused by by the the very very high high pressure pressure difference difference between between the the cylinder cylinder and and the the ambient ambient pressure. pressure. (2).(2 With). With the the increase increase of oil of inlet oil temperature, inlet temperature, the contact the timecontact between time pistonbetween and piston cavity and becomes cavity longerbecomes and longer longer. and longer. TheseThese results results show show that that whenwhen the pump pump is is operating, operating, the the eccentricity eccentricity appears appears to be to up be and up anddown downwith withthe changes the changes of piston of piston position position in the in distribution the distribution circle. circle.The bottom The bottom of piston of pistonmakes makesphysical physicalcontact contact with the with cylinder the cylinder during during a largea part large of part the of operating the operating time, there time, willthere be will more be more energy energyconsumption. consumption. AccordingAccording to to the the six six simulation simulation results, results, the the curve curve of contactof contact time time with with the the inlet inlet oil temperatureoil temperature is fittedis fitted out (seeout Figure(see Figure 11). In11). general, In general, there there should should be a positive be a positive correlation correlation between between contact contact time and time solidand friction. solid friction. Therefore, Therefore, the contact the contact time is time regarded is regarded as a simple as a simple measurement measurement of solid of friction,solid friction, the longerthe longer the contact the contact time, andtime, the and more the themore energy the energy consumption. consumption. From ourFrom analysis, our analysis, it is concluded it is concluded that thisthat relationship this relationship between between contact contact time and time inlet and oil inlet temperature oil temperature should should be expressed be expressed by Equation by Equation (35). The(35) values. The valuesA, B, C Aand, B, DC andare constants,D are constants, which which are determined are determined by the by material the material properties, properties, physics physics and operatingand operating conditions conditions of the of sliding the sliding pair pair and and lubricating lubricating oil andoil and can can be obtainedbe obtained by by the the simulation simulation methodmethod presented presented in in this this paper. paper. In thisIn this case, case,A = A 0.15, = 0.15,B = B 0.04, = 0.04,C = C− =84.37, −84.37, and andD D= 0.25.= 0.25. The The R-square R-square forfor the the fitting, fitting, analyzed analyzed by by MATLAB, MATLAB, is 0.97.is 0.97. t tc c ABTCD tanh  in   (35) t = A · tanh(B(Tin + C)) + D (35) ttotal total

Figure 11. Curve of contact time with oil inlet temperature. Figure 11. Curve of contact time with oil inlet temperature.

Figure 11 shows that the contact time increased with the increase of inlet oil temperature, and Figure 11 shows that the contact time increased with the increase of inlet oil temperature, and the the growth rate increased first and then decreased, but would not increase all the time. From◦ 20°C◦ to growth60°C, rate the increased contact time first andincrease thend decreased, slowly and but the would temperature not increase did all not the time. have From much 20 effectC to 60 on C, the the contact time increased slowly and the temperature did not have much effect on the movement of movement of the piston.◦ From◦ 60 °C to 100 °C, the contact time increased by a large margin, which themay piston. be due From to 60 theC decrease to 100 C, in the viscosity contact as time the increased temperature by a continueslarge margin, to increase, which may resulting be due into a thesignificant decrease indecrease viscosity in asthe the bearing temperature capacity continues of the lubricant to increase, film. resulting The piston in a is significant in continuous decrease contact in the bearing capacity of the lubricant film. The piston is in continuous contact with the cylinder under with the cylinder under the action of centrifugal◦ force. After 100 °C, the contact time between the thepiston action and of centrifugal the cylinder force. will After finally 100 reachC, the a balance contact time due tobetween the constraints the piston of and the the structure cylinder of willpiston/cylinder finally reach ainterface balance dueand tothe the working constraints conditions of the structure(pressure). of piston/cylinder interface and the working conditions (pressure).

Energies 2018, 11, 1842 15 of 16

6. Conclusions A numerical multi-discipline modeling method suitable for describing the lubricant oil film characteristics in piston pump has been developed. The results of velocity field and temperature field are accurate compared with the experiment. Considering the thermal effect, the quantitative relationship between oil inlet temperature and contact time between piston and cylinder is obtained in this paper, and it is shown that the relationship is a hyperbolic tangent. Thus, it can be predicted when and where the contact will appear under different temperature conditions. This method can be helpful when it refers to the design and analysis of a piston pump. In this paper, we consider the thermal effect on the oil film characteristics. In the future, we would include the oil pressure, axis rotate speed and the cavitation in the model and analyze all these factors’ effects on the pump operation performance, such as oil leakage and pressure fluctuation. We will also establish a test rig to verify the simulation results.

Author Contributions: Conceptualization, Yueheng Song and Jiming Ma; Methodology, Formal Analysis, Investigation and Writing-Original Draft Preparation, Yueheng Song; Writing-Review & Editing, Supervision and Project Administration, Shengkui Zeng. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Yang, H.Y.; Ding, F.; Yang, X.P.; Lu, Q. Hydraulic power systems for trunk line aircrafts. China Mech. Eng. 2009, 18, 2152–2159. 2. Cai, Z.B.; Zhou, Y.; Qu, J. Effect of oil temperature on tribological behavior of a lubricated steel–steel contact. Wear 2015, 332–333, 1158–1163. [CrossRef] 3. McCallon, H.; Yousif, F.; Lloyd, T. The analysis of thermal effects in a full journal bearing. J. Lubr. Technol. 1970, 92, 578–587. [CrossRef] 4. Ferron, J.; Frene, J.; Boncompain, R.A. Study of the thermohydrodynamic performance of a plain journal bearing comparison between theory and experiments. J. Lubr. Technol. 1983, 105, 422–428. [CrossRef] 5. Rohde, S.M.; Oh, K.P. A thermoelastohydrodynamic analysis of a finite slider bearing. J. Lubr. Technol. 1975, 97, 450–460. [CrossRef] 6. Mi, J.T.; Meng, Y.G. THD analysis of rolling piston and journal bearings in rotary compressors. Tribol. Trans. 2016, 59, 195–207. [CrossRef] 7. Cheng, F.; Ji, W. A velocity-slip model for analysis of the fluid film in the cavitation region of a journal bearing. Tribol. Trans. 2016, 97, 163–172. [CrossRef] 8. Lu, P.; Zheng, X.; Li, W.; Yang, P.; Huang, H.; Yu, Y. A thermal-fluid coupling numerical study on the characteristics of air-oil two phase flow and heat transfer in a micro UAV bearing chamber. RSC Adv. 2017, 7, 44598–44604. [CrossRef] 9. Braun, M.J.; Hannon, W.M. Cavitation formation and modelling for fluid film bearings: A review. J. Eng. Tribol. 2016, 224, 839–863. [CrossRef] 10. Chan, C.W.; Han, Y.F.; Wang, Z.; Wang, J.; Shi, F.; Wang, N.; Wang, Q.J. Exploration on a fast ehl computing technology for analyzing journal bearings with engineered surface textures. Tribol. Trans. 2014, 57, 206–215. [CrossRef] 11. Gels, S.; Murrenhoff, H. Simulation of the lubricating film between contoured piston and cylinder. Int. J. Fluid Power 2010, 11, 15–24. [CrossRef] 12. Ma, J.M.; Chen, J.; Li, J.; Li, Q.L.; Ren, C.Y. Wear analysis of swash plate/slipper pair of axis piston hydraulic pump. Tribolo. Int. 2015, 90, 467–472. [CrossRef] 13. Olems, L. Investigations of the temperature behaviour of the piston cylinder assembly in axial piston pumps. Int. J. Fluid Power 2000, 1, 27–39. [CrossRef] 14. Ivantysynova, M.; Huang, C.C.; Andreas, J.J. Determination of Gap Surface Temperature Distribution in Axial Piston Machines. In Proceedings of the ASME International Mechanical Engineering Congress and Exposition, Chicago, IL, USA, 5–10 November 2006. Energies 2018, 11, 1842 16 of 16

15. Kazama, T. Thermohydrodynamic lubrication model applicable to a slipper of swashplate type axial piston pumps and motors (Effects of operating conditions). Tribol. Online 2010, 5, 250–254. [CrossRef] 16. Pelosi, M.; Ivantysynova, M. The impact of axial piston machines mechanical parts constraint conditions on the thermo-elastohydrodynamic lubrication analysis of the fluid film interfaces. Int. J. Fluid Power 2013, 14, 35–51. [CrossRef] 17. Zecchi, M. A Novel Fluid Structure Interaction and Thermal Model to Predict the Cylinder block—Valve Plate Interface Performance in Swash Plate Type Axial Piston Machines. Ph.D. Thesis, Purdue University, West Lafayette, IN, USA, December 2013. 18. Wen, S.Z.; Huang, P. Principle of Tribology, 4th ed.; Tsinghua University Press: Beijing, China, 2012. 19. Khonsari, M.M.; Wang, S.H. On the fluid-solid interaction in reference to thermoelastohydrodynamic analysis of journal bearings. J. Tribol. 1991, 113, 398–404. [CrossRef] 20. Zhang, Z. Analysis of the Thermohydrodynamic Performance of Dynamically Loaded Bearings in Consideration of Non-Netonian, Derformation and Surface Topography. Ph.D Thesis, Shanghai Jiao Tong University, Shanghai, China, 2014. 21. Yang, S.M.; Tao, W.S. Heat Transfer, 4th ed.; Higher Education Press: Beijing, China, 2006. 22. Pelosi, M.; Ivantysynova, M. A Novel Thermal Model for the Piston/Cylinder Interface of Piston Machines. In Proceedings of the ASME Dynamic Systems and Control Conference, Hollywood, CA, USA, 12–14 October 2009.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).