Original Article Piston ring performance in two-stroke marine diesel engines: Effect of hydrophobicity and artificial surface texturing on power efficiency Proc IMechE Part J: J Engineering Tribology 0(0) 1–24 ! IMechE 2017 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1350650117736638 journals.sagepub.com/home/pij Eleftherios Koukoulopoulos and Christos I Papadopoulos Abstract In the present work, an algorithm for the solution of the Reynolds equation incorporating the Elrod–Adams cavitation model and appropriately modified to account for hydrophobic surfaces has been developed and solved by means of the finite difference method. The algorithm has been utilized to calculate the frictional characteristics of piston rings of a large two-stroke marine diesel engine, and to evaluate their performance, in terms of minimum film thickness, friction force, and power loss over a full-engine cycle, including time-dependent phenomena. For improving frictional behavior, two surface treatments of the piston ring surface have been studied, namely hydrophobicity and artificial surface texturing, which are introduced at appropriate parts of the ring face. Following a parametric analysis, optimal texturing and hydrophobicity design parameters have been identified for operation with maximum value of minimum film thickness and minimum friction losses. The present results demonstrate that substantial performance improvement can be achieved if hydrophobicity or artificial surface texturing is properly introduced at the faces of a piston ring. Keywords Piston ring, hydrodynamic lubrication, Reynolds, cavitation, Elrod–Adams, finite difference method, hydrophobicity, texturing, marine diesel engine Date received: 10 February 2017; accepted: 26 July 2017 Introduction Tribology studies friction, wear and lubrication of mechanical parts. Friction and wear may be substantially reduced when a thin layer (ﬁlm) of material, usually liquid, but also gas or solid, separates two sliding surfaces.1 Hydrodynamic lubrication has been a subject of extensive research in recent decades, applied to the study of mechanical components such as bearings, piston rings, seals, etc. The tribological system studied in the present paper is the piston ring—liner reciprocating mechanism of a large two-stroke marine diesel engine. Piston rings are circular metallic rings placed around the piston with a certain pretension; their main function is to isolate the combustion chamber volume with minimum friction. In particular, they aid in minimizing gas blow-by (leakage) from the combustion chamber to the crankcase. Pretension and gas pressure acting on the back face of piston rings sum up to the operational force of the piston ring face against the cylinder liner, giving rise to frictional forces and power loss. In large two-stroke marine diesel engines, friction losses of the piston ring assembly correspond to approximately 25% of the total engine friction losses.2 The tribological behavior of piston rings can be eﬃciently predicted on the basis of the Reynolds equation.3 A plurality of work has been reported in contemporary literature in the subject of piston ring tribology. An important study on piston rings has been published by Jeng in 1992,4 who solved the transient problem of hydrodynamic lubrication of piston rings with the use of the Reynolds equation. An extension of the basic modeling of Jeng4 to account for starved oil conditions at the ring inﬂow region has been published by the same author.5 A combined numerical and experimental work on piston ring friction in internal combustion engines has been National Technical University of Athens, School of Naval Architecture and Marine Engineering, Zografos, Greece Corresponding author: Eleftherios Koukoulopoulos, National Technical University of Athens, School of Naval Architecture and Marine Engineering, Heroon Polytechniou 9, Athens 157 80, Greece. Email: [email protected] 2 presented by Wakuri et al.,6 whereas Livanos and Kyrtatos have addressed the problem of predicting friction losses of the piston assembly components (piston rings, piston skirts, and gudgeon pin) of four-stroke marine diesel engines operating at constant and variable engine rotational speed.7 Regarding lubricant cavitation in hydrodynamic lubrication, the simplest approach is to alter the full ﬁlm pressure results, by setting all predicted negative pressures equal to the cavitation pressure (halfSommerfeld type of boundary conditions). Although this assumption produces reasonable load values, continuity of pressure derivative at the cavitation boundary, as well as mass conservation in the divergent ring region is violated. The former may be tackled by assuming that the pressure derivative is zero at the ﬁlm rupture boundary (Reynolds boundary condition); this boundary condition has been used extensively in the literature. It gives accurate results regarding lubricant pressure, and it is capable of identifying correctly the ﬂuid rupture boundary. However, mass conservation of lubricant in the cavitating region is violated and ﬁlm reformation boundary is poorly predicted.8 To handle these deﬁciencies, several algorithms have been developed, the most popular one being the Elrod–Adams cavitation model.9,10 The Elrod–Adams algorithm was reﬁned by Vijayaraghavan and Keith,11 by implementing a half-step ﬁnite diﬀerence scheme for the shear term of the Reynolds equation and by adopting the switch function proposed by Elrod and Adams so as to identify the cavitation region and vanish the pressure terms there. The original switch function utilized a variable cavitation index, which took the value of either 0 or 1 at every node of the domain, corresponding to cavitating or full ﬁlm region, respectively. This variable has been reported by several researchers to cause numerical oscillations and instabilities, due to its abrupt change (0 and 1), while trying to predict the cavitation boundary. Khonsari and Fesanghary12 proposed a modiﬁcation of the original switch function, which smooths the transition between the full ﬁlm and the cavitating region and accelerates the convergence speed. The present work builds on top of recent research related to the eﬀect of surface treatment methods on the tribological performance of hydrodynamically lubricated contacts. In particular, two diﬀerent methods will be evaluated, those of hydrophobicity and artiﬁcial surface texturing, which have been applied in journal and thrust bearings giving substantial potential of performance improvement, as it can also be seen in Fatu et al.,13 Pavlioglou et al.,14 and Guo-Jun et al.15 and Etsion et al.,16 Papadopoulos et al.,17 and Xiong and Wang18 respectively. Hydrophobic surfaces draw their origins from the lotus leaf, and, due to their particular nanostructure, they are wetting resistant, exhibiting low levels of friction during ﬂuid ﬂow. In order to modify Reynolds Proc IMechE Part J: J Engineering Tribology 0(0) equation so as to account for hydrophobic surfaces, the no-slip boundary condition on the ﬂuid-wall interface should be replaced with proper slip boundary conditions, see Fatu et al.13 and Guo-Jun et al.15 Instead of using Reynolds equation, Computational Fluid Mechanics (CFD) simulations can also be used, like in Pavlioglou et al.,14 and it has been conﬁrmed that the corresponding results match suﬃciently. Artiﬁcial surface texturing refers to the introduction of small periodic irregularities of diﬀerent shapes on a surface, in the form of small dimples coming in rectangular, trapezoidal, cylindrical, or spherical shape. Their potential of increasing load-carrying capacity and reducing frictional coeﬃcient has been veriﬁed by many studies, such as those presented in Etsion et al.16 and Papadopoulos et al.17 It must be noted that cavitation is more likely to occur on textured surfaces, because of the textures creating steep divergent and convergent geometries. Therefore, implementation of a cavitation model is recommended, if accurate simulation results are sought. In the present study, the eﬀect of surface treatment (in the form of hydrophobicity or artiﬁcial surface texturing) on the performance characteristics of the ﬁrst (compression) piston ring of a two-stroke marine diesel engine is investigated. First, a parametric analysis is performed to determine the optimal plain proﬁle of the piston ring. Then, hydrophobicity or artiﬁcial surface texturing is introduced in parts of the ring face. Ring performance is evaluated on the basis of minimum ﬁlm thickness and friction power losses over a full engine cycle. The results demonstrate a substantial potential of performance improvement for both surface treatment technologies. The present paper is organized as follows: the geometric details of the study piston ring are ﬁrst outlined, followed by the presentation of the governing equation and the computational approach. Subsequently, the computational results are presented and discussed, and ﬁnally, the main conclusion is summarized. Problem setup Geometry The piston ring—liner is a reciprocating tribological system analogous to a slider bearing, where the ring is the rotor and the liner is the stator. While the piston moves along the cylinder, the piston ring undergoes a complex motion consisting of a sliding motion parallel to the liner surface and a squeeze motion perpendicular to the liner surface. The piston ring geometry can be simpliﬁed by considering only a 2-D vertical section, as seen in Figure 1. The lubricant ﬂuid ﬁlls the gap between the piston ring and the liner, so ﬂuid ﬁlm geometry is a function of x-coordinate, h(x). Film thickness, h(x), can be expressed as h(x) ¼ hs(x) þ hmin, where hs(x) is the piston ring face proﬁle and hmin is minimum ﬁlm Koukoulopoulos and Papadopoulos 3 Figure 1. Sketch of the cross section of a typical piston ring. thickness (the minimum distance between the ring and the liner). Minimum ﬁlm thickness varies during a full engine cycle, therefore h is also a function of time (h ¼ h(x,t)). A parabolic function is usually suitable to describe the piston ring face proﬁle after running-in wear, therefore hs(x) can be expressed as hs ðxÞ ¼ c b 2 þo 2 2 ðx oÞ ð1Þ where, b is the piston ring width, c is the crown height, and o the oﬀset of the crown from the center point of the piston ring width (see Figure 1). It is assumed that the piston ring moves with a positive velocity, U > 0, during the downstroke motion and with a negative velocity, U < 0, during the upstroke motion. Governing equation The piston ring—liner reciprocating system is assumed to operate in the regime of hydrodynamic lubrication, which is governed by the Reynolds equation. The following assumptions are made for the present model; (a) the tilting motion as well as any elastic deformations of the ring are neglected and (b) isothermal ﬂow conditions are assumed (temperature is assumed constant and equal to a properly selected mean value, depending on operating conditions). At ﬁrst, Reynolds equation is modiﬁed to account for hydrophobic surfaces, by implementing appropriate boundary conditions at the lubricant-ring interface, as also presented in Fatu et al.13 In particular, lubricant is allowed to slip on the hydrophobic boundary when the local shear stress value is higher than a critical shear stress value, which is speciﬁc property of each hydrophobic surface. In the present work, the critical shear stress value is assumed zero, meaning that any nonzero value of shear stress can lead to lubricant slip over the hydrophobic boundary. When slip occurs the ﬂuid slips with a velocity proportional to shear stress multiplied by a factor called slip coeﬃcient, al for hydrophobic liner surface and ar for hydrophobic ring surface. The corresponding wall boundary conditions for ﬂuid velocity are hereinafter presented uz¼0 @u ¼ U al C,l @z z¼0 @u uz¼h ¼ ar C,r @z z¼h ð2Þ ð3Þ where c,l and c,r are the values of critical shear stress at the liner and ring surfaces, respectively. The ﬁnal form of the Reynold equation for the pressure region of the ring is the following @ h2 @p h2 þ 4ðal þ ar Þh þ 122 al ar @x 12 @x h þ ðal þ ar Þ 2 2 @ @p h h þ 4hðar þ al Þ þ 12al ar 2 þ @y @y 12 h þ ðar þ al Þ 2 U @ h þ 2ar h ar @h ¼ U 2 @x h þ ðal þ ar Þ h þ ðal þ ar Þ @x h ar h þ 2al ar 2 @h @p @p @h þ þ 2 h þ ðal þ ar Þ @x @x @y @y @ 1 h2 al C,l ar C,r þ 2al ar h C,l C,r þ @x 2 h þ ðal þ ar Þ ar al C,r C,l þ ar C,r h @h þ h þ ðal þ ar Þ @x @ h ar C,r þ al C,l h þ 2ar al C,r þ C,l @y 2 h þ ðar þ al Þ ar C,r h þ ar al C,r þ C,l @h @h þ þ @y @t h þ ðar þ al Þ ð4Þ where is the lubricant viscosity, h is the ﬂuid ﬁlm thickness, and p is the pressure for which the equations are solved. The detailed derivation of equation (4) is presented in Appendix 1. 4 Proc IMechE Part J: J Engineering Tribology 0(0) The lubricant oil density is related to the ﬁlm pressure through the bulk modulus deﬁnition Boundary conditions and cavitation modeling The piston ring operation depends on the values of the upstream and downstream pressure, respectively. For the ﬁrst compression ring of the piston, studied in the present work, the pressure at the top edge of the ring is considered equal to the combustion chamber pressure, pch, whereas the pressure at the bottom edge (i.e., the pressure between the ﬁrst and the second compression rings) p1–2 is assumed to be half of the combustion chamber pressure (p1–2 ¼ pch/2). At the inlet of the piston ring, fully ﬂooded conditions have been assumed, meaning that oil exists in abundance at the inlet region. The leading and trailing pressure of the ring is determined at each time step, depending on the direction of motion; during downstroke, p1–2 is the leading pressure and pch the trailing pressure, whereas at upstroke, the two pressure values are swapped. Because of the converging–diverging geometry of the piston ring proﬁle, cavitation always occurs at the diverging part. A simple way to model cavitation is with the use of the Reynolds boundary condition, which sets negative pressures in the lubricant domain equal to zero, and ensures that the pressure gradient at the transition boundary between the active and the cavitating region is also zero. However, this boundary condition violates mass conservation in the cavitating region, it is incapable of predicting the reformation boundary and gives practically zero information about the ﬂuid condition in the cavitation area. Therefore, in the present work, the well-known Elrod–Adams cavitation algorithm10 has been implemented. In particular, the lubricant domain is divided into two zones. The ﬁrst one is the pressurized (active) zone, where the ﬁlm is fully developed, and the Reynolds equation applies. The second is the cavitation (passive) zone, where only a fraction of the gap is occupied with oil, therefore ﬁnger-like striations of liquid and gas are observed. Here, a universal equation for both the active and the passive zone is used, containing the fractional ﬁlm content variable . In the active zone, the mass content per unit ﬁlm area is equal to ch, where c is the lubricant oil density at cavitation pressure pc and h is the ﬁlm thickness. In the passive zone, the lubricant density is constant and equal to c, but the mass content is now ch. In the active zone, density varies due to the pressure variation, meaning that the oil is considered slightly compressible and the mass content is higher than that corresponding to cavitation pressure pc. Consequently, the fractional ﬁlm content is equal to /c. The two-dimensional Reynolds equation can be written as follows, and it fully applies to the full ﬁlm region @ h3 @p @ h3 @p U @ðhÞ @ðhÞ þ þ ¼ @x 12 @x @y 12 @y 2 @x @t ð5Þ ¼ @p @ ð6Þ The idea of this methodology is to substitute pressure p with a universal variable , which is equal to the ratio of density along the ﬁlm area and the density at the cavitation pressure. Variable takes values slightly above unity in the full ﬁlm region, because of the relative compression of the lubricant as explained above, while in the cavitation region, takes values below unity. ¼ ¼ C 4 1 in the full film region ð7Þ 5 1 in the cavitation region Due to the fact that the ﬁlm pressure is constant in the cavitation region, a switch function is introduced in the pressure–density relation, in order to exclude the pressure terms in the cavitation region, and keep only the Couette ﬂow term @p @p g¼ g ¼ ¼ @ @ 1 in the full film region 0 in the cavitation region ð8Þ Directly integrating equation (8), the following expression is obtained p ¼ pC þ g ln ð9Þ Replacing equation (9) into equation (5) yields @ gðÞh3 @ @ gðÞh3 @ U @ðhÞ @ðhÞ þ þ ¼ @x @y 2 @x @t 12 @x 12 @y ð10Þ The ﬁnal utilized global equation is derived from the combination of the cavitation model (equation (10)) and the modiﬁed Reynolds equation for slip boundary conditions (equation (4)), setting the critical shear stress values equal to zero @ h3 h2 þ 4hðaS þ ah Þ þ 122 aS ah @ C gðÞ @x @x 12 hðh þ ðaS þ ah ÞÞ @ h3 h2 þ 4hðaS þ ah Þ þ 122 aS ah @ C gðÞ þ @y @y 12 hðh þ ðaS þ ah ÞÞ U @ h2 þ 2hah h hah þ 22 aS ah C ¼ þ C 2 @x 2 h þ ðaS þ ah Þ h þ ðaS þ ah Þ @ @h @ @h þ gðÞ gðÞ @x @x @y @y C ah @h @ ðC hÞ þ U @t h þ ðaS þ ah Þ @x ð11Þ Koukoulopoulos and Papadopoulos 5 Unsteady solution and equilibrium condition Solving the problem of hydrodynamic lubrication of a piston ring over an entire engine cycle demands equilibrium between the external forces acting on the ring and the hydrodynamic forces in the lubricant domain at each time step. In particular, equilibrium of the ring is attained when external forces due to the ring pretension and gas pressure acting on the ring are balanced by hydrodynamic forces developed in the lubricant ﬁlm, separating the ring from the liner. At high external loads, the ring will move closer to the liner, decreasing hmin, whereas at low external loads, the ring will move in the opposite direction, increasing hmin. Therefore, because velocity varies along the piston stroke and ring load is time dependent, hmin will also be a function of time. At very low values of piston speed, the ring is prone to come in contact with the liner, as the ﬂuid speed is not adequate to generate the hydrodynamic wedge necessary for lubrication. In order to model this phenomenon, a threshold value of hmin has been selected. Here, this value is considered equal to the composite roughness height, r, of the ring face; lower values of hmin are set equal to r. In this case, a constant value of friction coeﬃcient, c, corresponding to dry friction, is utilized. As already mentioned, the external force at each time step is the sum of the elastic pretention force of the ring and the gas force acting on the back face of the ring Fext ¼ Pel þ Pbk Pel ¼ 2T b bB Pbk ¼ maxðpch , p12 Þ b ð12Þ ð13Þ ð14Þ where Pbk is the ring elastic pressure and T the tangential ring pre-tension force. Here, Pbk is calculated by assuming that at the back face of the piston ring, the acting pressure is the maximum value between pressures pch and p1–2. Integral quantities At every time step, the external ring force should be balanced by the hydrodynamic force developed in the lubricant domain. This force, called the load-carrying capacity of the lubricant can be calculated as Z lZ W¼ b pdxdy 0 ð15Þ 0 where p is the lubricant pressure obtained from equation (9) after the solution of equation (11), whereas b and l are the piston ring width and the arc length along the perimeter, respectively. Friction force, F can be calculated as follows Z lZ b h dp U dx dy F¼ 2 dx h 0 0 ð16Þ In the cavitation domain of the ﬂuid area, only a portion of the region is ﬁlled with ﬂuid, and this portion in our paper is represented by the fractional ﬁlm content variable ‘‘.’’ Therefore, the equation calculating the friction in the cavitation region, is accordingly modiﬁed, integrating the variable ‘‘,’’ as only this fraction of the ﬂuid causes friction. The according equation is presented below F¼ Z lZ b h dp U dx dy 2 dx h 0 0 ð17Þ When the piston ring is in contact with the liner, in other words hmin is equal to composite roughness height r, the value given to the instantaneous friction force is the vertical load multiplied by the dry friction coeﬃcient, c, here being equal to 0.08. The total power loss, PL, of the ring at a certain time step can be calculated as PL ¼ F U ð18Þ At each time step, the value of hmin for which force equilibrium of the ring is reached is calculated by means of a second order Newton–Raphson method, also called Halley’s method. Equilibrium is reached when Wðhmin Þ Fext ¼ 0 where W(hmin) is the hydrodynamic force acting on the ring at a value of minimum ﬁlm thickness hmin, and Fext is the external ring force. Validation of the algorithm First, the results of the present algorithm for solving the Reynolds equation algorithm are validated by comparing the calculated performance characteristics of a piston ring with those published by Jeng.4 In particular, the ﬁrst compression ring of a fourstroke diesel engine is considered, and the results are presented in Figure 2, exhibiting a very good agreement. At this point it must be clariﬁed that the squeeze ﬁlm motion has been included in the solution of the transient problem, in the means of including the terms dh/dt. Next, the developed solution algorithms of the Reynolds equation using the Reynolds or the Elrod– Adams mass conservation model are validated against the results of Giacopini et al.8 and Guo-Jun et al.15 In particular, three diﬀerent cases, corresponding to (a) a 6 Proc IMechE Part J: J Engineering Tribology 0(0) Figure 2. (a) Minimum film thickness, (b) power loss, and (c) friction force against crank angle: Comparison between the present results and those of Jeng.4 diverging–converging sinusoidal slider, (b) a textured slider, and (c) a slider with hydrophobicity at part of the stator have been considered. In Figure 3, pressure proﬁles and integral quantities are presented in comparison to literature results. A very good agreement is observed for all the studied cases. Computational results Computations were performed for the ﬁrst compression ring of a piston of a large two-stroke marine diesel engine. The reference engine in this work is similar to the RT-ﬂex58T-B engine by Wärtsilä, which at maximum continuous rating (MCR) delivers 2125 kW per cylinder at 105 r/min. Figure 4(a) shows the piston speed against crank angle (CA) for two diﬀerent values of engine rotational speed, N, namely those of 105 r/min and 66.1 r/min. As mentioned earlier, the top edge of the reference ring is assumed to be at a pressure equal to the combustion pressure of the engine, whereas pressure at the bottom edge of the ring is assumed to be half of the combustion pressure. Figure 4(b) presents the combustion chamber pressure distribution at the nominal load (100% of MCR, 105 r/min) and at a low load (25% of MCR, 66.1 r/min). The principal characteristics of the engine and the reference piston ring design are summarized in Table 1; a typical sketch of the piston ring face proﬁle has already been presented in Figure 1. All simulations of the present study are carried out over two consecutive engine cycles (720 of CA). It is noted that at the beginning of the ﬁrst cycle, the gradients of the problem variables are not a priori known, therefore the results are not accurate. This problem is resolved at the beginning of the second engine cycle, therefore, in this paper, the presented results correspond to the second computed engine cycle. Mesh study Before proceeding to the ﬁnal calculations, it is necessary to select the appropriate number of nodes of the computational model. In Figure 5, a mesh study is presented. In particular, minimum ﬁlm thickness, friction force, and maximum pressure are plotted against node number, for the time step corresponding to a CA of 90 . Based on Figure 5, a grid size of 201 nodes in the x-direction of Figure 1 is selected for the computations of the present study. Effect of time derivatives (squeeze film motion) While most publications utilizing the Elrod–Adams mass conservation model solve the steady-state Koukoulopoulos and Papadopoulos 7 Figure 3. Pressure profiles calculated using the Elrod–Adams cavitation algorithm for (a) a diverging–converging sinusoidal slider and (b) a simple textured slider, and comparison with the results of Giacopini et al8 Nondimensional values of load capacity (c) and friction force (d) against convergence ratio (k ¼ h1/h0-1) for a slider with hydrophobicity at part of the stator, and comparison with the results of Guo et al.15 Figure 4. (a) Piston speed versus crank angle at 105 r/min (100% engine load) and 66.1 r/min (25% engine load) and (b) pressure in the combustion chamber versus crank angle for engine loads of 100% and 25%. problem, in the present work, the full time-dependent problem has been considered. The main diﬀerence between the two approaches, is the inclusion in the latter of the squeeze ﬁlm term, by which the time derivative of ﬁlm thickness is taken into consideration. A detailed analysis of the squeeze eﬀect on the resulting ﬁlm thickness behavior can be seen in Taylor,19 which comes in perfect agreement with the results presented hereinafter. In Figure 6, the pressure and ﬁlm thickness distribution at the CA of 179 is presented 8 Proc IMechE Part J: J Engineering Tribology 0(0) for both calculation approaches. This CA value has been selected because the piston is located close to the bottom dead center (BDC) and piston speed is almost equal to zero. With zero piston speed, no pressure ﬁeld can be generated due to hydrodynamic lubrication, as there is no sliding motion. According to Figure 6, if time derivatives are not taken into account, the piston ring is practically in contact with the liner (ﬁlm thickness is almost equal to composite roughness r, see thick dotted line of Figure 6). In addition, the pressure distribution is very steep (thick continuous line of Figure 6), because of the generated steep hydrodynamic wedge, which also leads to a large cavitation region, being almost 50% of the total piston ring area. On the other hand, when time Table 1. Geometric and operational characteristics of the studied engine. Engine parameters derivatives are taken into consideration, due to the squeeze ﬁlm term dh/dt, pressure ﬁeld is developed (thin continuous line of Figure 6) capable of separating the two surfaces with a thicker ﬁlm, decreasing the possibility of metal-to-metal contact and wear. For both cases, the resulted load is the same, as integrating the two pressure curves they will yield the same result. The squeeze ﬁlm eﬀect has been taken into consideration for all the simulations of this paper. As discussed earlier, this inclusion aﬀects substantially the ring performance, mainly in the regions of the top dead center (TDC) and BDCs, where very low values of piston speed are present (see e.g., the results presented in Figure 8(a), in the CA regions of 0 and 180 ). Reference (plain) piston ring design, nominal operating condition (100% engine load, 105 r/min) At ﬁrst, the reference piston ring design of Table 1 is considered. Simulations are performed for the Bore, B 580 mm Stroke, S 2416 mm Connecting rod length, R 2241 mm Revolution speed, N 105 r/min load 100%, 2125 kW/cylinder 66.1 r/min load 25%, 531.25 kW/cylinder Oil dynamic viscosity, 0.19 Pa.s First (top) compression piston ring (see Figure 1) Radial width Ring thickness, b Crown height, c Offset, o Composite roughness, r Pre-tension force, T Dry friction coefficient, c 28 mm 16 mm 3 mm 0 mm 0.2 mm 49,744 N 0.08 Figure 5. Mesh study of the plain piston ring at 100% engine load. Figure 6. Pressure and film thickness distribution at CA ¼ 179 (approximately zero sliding velocity). Effect of the squeeze film term on the solution. Koukoulopoulos and Papadopoulos 9 Figure 7. Reference piston ring design, 100% engine load, 105 r/min: Operational indices versus engine crank angle. Figure 8. One hundred percent engine load, 105 r/min (a) Minimum film thickness, and (b) power loss, against crank angle, for different crown height values. nominal operating conditions of the diesel engine, corresponding to 100% of load and to 105 r/min rotational speed. In Figure 7, several performance parameters of the ring, such as minimum ﬁlm thickness, power loss, maximum pressure, and cavitation area are plotted against CA. Minimum ﬁlm thickness ranges from 5 mm to 12 mm approximately. The maximum pressure acting on the back surface of the 10 Proc IMechE Part J: J Engineering Tribology 0(0) piston ring surface is observed at 11.5 of CA, where the combustion phenomenon is intense. Friction force and friction coeﬃcient exhibit their maximum values at values of CA equal to 64.6 and 229.9 , where piston speed is maximum, and their minimum values at CA ¼ 0 and CA ¼ 179.8 , where piston speed is zero. In the following paragraphs, a brief parametric analysis will be presented, aiming at identifying the eﬀect of the main piston ring design parameters (crown height, oﬀset) on its tribological performance. assumed equal to 3 mm; here values between 1 mm and 20 mm are also considered. Higher values of crown height lead to increased minimum ﬁlm thickness and decreased power loss at regions of high piston velocity. However, at regions of low piston velocity (near the TDC and BDC), minimum ﬁlm thickness is decreased, in comparison to the reference design, increasing the likelihood of asperity contact between the ring and the liner wall. Further, values of crown height higher than 12 mm will provide negligible additional beneﬁt. Effect of crown height Effect of offset Figure 8 shows the eﬀect of diﬀerent crown height values on the performance indices of the piston ring. The crown height of the reference piston ring is In Figures 9 and 10, the eﬀect of diﬀerent oﬀset values on the operational indices of the piston ring is presented. The reference piston ring has zero oﬀset, Figure 9. One hundred percent engine load, 105 r/min (a) Minimum film thickness, and (b) power loss, against crank angle, for different positive offset values. Figure 10. One hundred percent engine load, 105 r/min (a) Minimum film thickness, and (b) power loss, against crank angle, for different negative offset values. Koukoulopoulos and Papadopoulos 11 o ¼ 0 mm; here, oﬀset values of 0.8 mm, 1.6 mm, 3.2 mm, 0.8 mm, 1.6 mm, 3.2 mm are considered. Positive and negative oﬀset values are deﬁned as shown in Figure 1. Based on Figure 9, we observe that positive values of oﬀset, o, lead to decreased minimum ﬁlm thickness at the ﬁrst half of the engine cycle (0 –180 of CA), and to increased minimum ﬁlm thickness at the second half of the engine cycle. At CA values close to 180 , where piston speed is minimum, positive values of oﬀset lead to decreased values of minimum ﬁlm thickness, which increases the likelihood of asperity contact between the ring and then liner. Power loss increases substantially at the ﬁrst half of the engine cycle, whereas, at the second half, it exhibits a small decrease, in comparison to the reference case. On the other hand, negative values of oﬀset, o, lead to increased minimum ﬁlm thickness at the ﬁrst half of the engine cycle and to a less pronounced decrease of minimum ﬁlm thickness at the second half of the engine cycle. The overall minimum ﬁlm thickness remains at the levels of that corresponding to the reference case, except for the case of o ¼ 3.2 mm where asperity contact is observed at CA ¼ 180 . Power loss exhibits substantial reduction at the ﬁrst half of the engine cycle, whereas it is slightly increased at the second half of the engine cycle. Based on Figures 9 and 10, a zero oﬀset ring provides the best performance for the engine of the present study. Hydrophobic piston ring surface The ﬁrst studied conﬁguration is that of a piston ring with hydrophobic properties at parts of its surface. In particular, two regions are selected for application of hydrophobicity, lying at either side of the ring surface. This design features an additional practical advantage, as the converging and diverging parts of the ring come in contact with the liner less frequently than the mid part, reducing the potential wear of the hydrophobic surface. To characterize the hydrophobicity of a surface, the concept of slip length bs, is commonly used. Slip length of a super-hydrophobic surface is the ﬁctitious distance below the surface at which ﬂuid velocity extrapolates to zero. Here, the nondimensional slip length, b*, deﬁned as the ratio of slip length to a characteristic value of ﬁlm thickness, is utilized for characterizing super-hydrophobic surfaces. In the present work, state equations are solved over a full engine cycle, therefore minimum ﬁlm thickness diﬀers from time-step to time-step, making necessary the use of a ﬁxed value of ﬁlm thickness as a characteristic value of ﬁlm thickness; here, composite roughness, r, is selected for this purpose. Therefore, b* is deﬁned here as b ¼ brs . The location and extent of the piston ring hydrophobic regions are controlled by the nondimensional parameters SBs, SBe, STs, STe, as Figure 11. Piston ring face profile with the regions of hydrophobicity highlighted; the four dots mark the start and end of each hydrophobic region. explained in Figure 11. A description of those parameters and their initial values is presented in Table 2. Contemporary technology allows fabrication of super-hydrophobic surfaces exhibiting a slip length ranging from a few hundred nanometers up to 50 mm (as presented in Rothstein,20 Neto et al.,21 Tsai et al.,22 Verho et al.23 and references therein). Nevertheless, manufacturing of durable hydrophobic surfaces continues to be a challenge, because the eﬀect of hydrophobicity is hindered by the sensitivity of the micro-roughness, which is responsible for the hydrophobic behavior of the surface. In more detail, mechanical wear and operation under pressure of the hydrophobic surface, will lead to gradual deterioration of the non-wettability properties, which is translated in a decrease of the active slip-length.23 Therefore, for the computations of the present section, a slip length value bs ¼ 20 mm is assumed, which is an achievable value, according to the previous analysis and corresponding to a nondimensional value of b* ¼ 100 (the composite roughness r of the ring face here is equal to 0.2 mm). At ﬁrst, the initial design of a hydrophobic ring, presented in Table 2 is considered. In Figure 12, 12 Proc IMechE Part J: J Engineering Tribology 0(0) Table 2. Piston ring with hydrophobicity. Parameters controlling the location and extent of the hydrophobic regions of Figure 11 and their initial values. Variable Description SBs Bottom hydrophobic region: Nondimensional x coordinate of slip start location Bottom hydrophobic region: Nondimensional x coordinate of slip end location Top hydrophobic region: Nondimensional x coordinate of slip start location Top hydrophobic region: Nondimensional x coordinate of slip end location SBe STs STe Initial value 0.1 0.3 0.7 a comparison between the tribological behavior of the conventional and the hydrophobic piston rings is presented. The introduction of hydrophobicity is proven beneﬁcial for the ring. In particular, minimum ﬁlm thickness is increased, corresponding to higher load capacity, whereas power loss is substantially decreased. To identify the optimum extent of both slip regions of the ring, a parametric analysis is performed, keeping unaltered the remaining design parameters of Table 2. Bottom slip region 0.9 In this section, the starting and end locations of the bottom hydrophobic region are varied around their initial values. In particular, in Figures 13 and 14, Figure 12. One hundred percent engine load, 105 r/min: (a) Minimum film thickness, and (b) power loss against crank angle. Comparison between the conventional and the hydrophobic piston ring designs. Figure 13. One hundred percent engine load, 105 r/min: (a) Minimum film thickness and (b) power loss, against crank angle, for different values of parameter SBs (start of bottom slip region). Koukoulopoulos and Papadopoulos 13 Figure 14. One hundred percent engine load, 105 r/min: (a) Minimum film thickness, (b) power loss, against crank angle, for different values of the end of bottom slip region, SBe. Figure 15. One hundred percent engine load, 105 r/min: (a) Minimum film thickness, and (b) power loss, against crank angle, for different values of start of top slip region, STs. minimum ﬁlm thickness and power loss are presented against CA, for diﬀerent values of SBs and SBe respectively. Following the results of Figures 13 and 14, altering the starting location of the bottom slip region aﬀects ring performance only during the downstroke motion of the piston, because during the upstroke motion, this area is mostly located in the cavitation region. On the other hand, the end of the bottom slip region aﬀects the operation of the ring during the whole cycle because this area of the ring is mostly located in the pressure area of the ring and determines the way the pressure is developed. Combining the results of both ﬁgures, the optimum design of the bottom slip region would be the one that extents from 0% to 30% of the piston ring surface. values. In particular, in Figures 15 and 16, minimum ﬁlm thickness and power loss are presented against CA, for diﬀerent values of STs and STe, respectively. Figures 15 and 16 reveal that the trend of minimum ﬁlm thickness and power loss is exactly the opposite from that corresponding to variation of design parameters of the bottom slip region. In particular, changing the starting point of the top slip location, aﬀects the ring operation throughout the whole engine cycle, as this area is always located in the region of the ring where pressure is developed, while altering the end of the top slip area aﬀects only the upstroke motion. Merging the results of both ﬁgures, the optimum design of the top slip region would be the one that extents from 70% to 100% of the piston ring surface. Top slip region Optimal piston ring design with hydrophobicity In this section, the starting and end points of the top hydrophobic region are varied around their initial Based on the outcome of the above parametric study, optimal behavior of the piston ring is attained by 14 Proc IMechE Part J: J Engineering Tribology 0(0) Figure 16. One hundred percent engine load, 105 r/min: (a) Minimum film thickness, (b) power loss, against crank angle, for different values of the end of top slip region, STe. Figure 17. One hundred percent engine load, 105 r/min (a) Minimum film thickness and (b) power loss, against crank angle: Comparison between the reference conventional (non-hydrophobic) piston ring design and the optimal hydrophobic design. introducing hydrophobicity in two regions, one at the bottom part of the ring extending from 0% to 30% of the ring surface and on at the top part extending from 70% to 100%. In Figure 17, the beneﬁcial eﬀect of hydrophobicity on the operation of the ring is depicted; the average minimum ﬁlm thickness, hmin over a full engine cycle is increased by 53%, whereas total friction power over a full circle is decreased by 56%. In order to examine if the above design remains beneﬁcial at lower loads and values of rotational speed, Figure 18 is generated, presenting the behavior of the optimal hydrophobic piston ring design, which resulted from the 100% engine MCR and 105 r/min rotational speed (nominal operating condition of the engine), operating now at 25% engine MCR and 66.1 r/min. This new operating condition is representative of engine operation of large cargo-carrying ships traveling at slow-steaming speeds to reduce fuel consumption. Based on the results of Figure 18, it can be observed that there is substantial performance improvement, even at lower loads, but less pronounced in comparison with that corresponding to nominal engine operation (as presented in Figure 17). Nondimensional slip length In order to study the theoretical limits of the technology of hydrophobicity, the determining parameter for the behavior of the hydrophobic surface, which is the nondimensional slip length, b* has been tested in a wide range of values. In order to clarify the eﬀect of diﬀerent values of b* a parametric study was performed in the range of 0–10,000, for the case of the optimal hydrophobic design of the piston ring. The resulting performance indices of the piston ring Koukoulopoulos and Papadopoulos 15 Figure 18. Twenty-five percent engine load, 66.1 r/min (a) Minimum film thickness and (b) power loss, against crank angle: Comparison between the reference conventional (non-hydrophobic) piston ring design and the optimal hydrophobic design as resulted for 100% load. Figure 19. One hundred percent engine load, 105 r/min (a) Minimum film thickness, and (b) power loss, against crank angle, for different values of non-dimensional slip length, b*. are presented in Figure 19. The corresponding average values of minimum ﬁlm thickness and power loss, as a function of b* are presented in Figure 20. The most substantial improvement is observed in the interval [10, 100] of b*, explaining also the selection of the reference value b* ¼ 100 for the computations of the present study. Further increase of b* above 1000 will lead to negligible additional performance improvement of the piston ring performance. Artificial texturing The second surface treatment technology that has been studied in the context of the present work is that of artiﬁcial surface texturing. It has been experimentally demonstrated that, with the use of contemporary technology, it is feasible to manufacture high resolution artiﬁcially textured surfaces, with the most common technique being that of laser surface texturing (LST), which has been successfully applied to surfaces of journal/thrust bearings, piston rings, and mechanical seals.24,25 Additional methods of generating high resolution texturing on surfaces of mechanical components, in particular micro-stereolithography, chemical etching, surface indentation, micromachining, lithography, electroplating and moulding (LIGA) processes, and laser ablation, have enabled the implementation of artiﬁcial texture patterns in machine components, with resolution accuracy of a few microns or even less. Similar to hydrophobicity, artiﬁcial surface texturing is introduced in two regions at the bottom and top parts of the piston ring surface. The location and extent of those regions are controlled by the nondimensional parameters TBs, TBe, TTs, and TTe. A typical sketch of the piston ring with the two textured regions is shown in Figure 21. 16 Proc IMechE Part J: J Engineering Tribology 0(0) Table 3. Piston ring with artificial surface texturing: Parameters controlling the location and extent of the textured regions and the geometry of the dimples of Figure 21. Variable Description Nd S Number of dimples Density of the dimple cell within the dimple cell Dimple depth Bottom textured region: Nondimensional x coordinate of texturing start location Bottom textured region: Nondimensional x coordinate of texturing end location Top textured region: Nondimensional x coordinate of texturing start location Top textured region: Nondimensional x coordinate of texturing end location dd TBs TBe Figure 20. One hunderd percent engine load, 105 r/min, average minimum film thickness and friction power of a full engine cycle as a function of nondimensional slip length b*. TTs TTe Initial values 3 0.5 5 mm 0.1 0.3 0.7 0.9 Figure 21. The initial values of TBs, TBe, TTs, TTe, Nd, S , and dd are given in Table 3. A comparison of performance parameters of the plain piston ring with those of the textured ring with the design parameters of Table 3, can be seen in Figure 22. The introduction of artiﬁcial surface texturing is proven beneﬁcial for the ring, leading to increased minimum ﬁlm thickness and substantially decreased power loss. To identify optimal design of the textured regions, a parametric analysis is performed for the basic design parameters TSs, TBe, TTs, TTe, and dd. Bottom textured region Figure 21. Face profile of a textured piston ring; each pair of dots marks the start and end of one texture cell. Once those parameters are deﬁned, the length of each textured area can be calculated. Next, this length is divided by the selected number of dimples, Nd, yielding the length of each dimple cell, lc. Finally, multiplying the length of dimple cell with the dimple density, T, yields the length of each dimple, ld. All dimples are centered in the corresponding dimple cell and bear the same depth dd. The corresponding texture geometry is also presented in In this section, the start and end locations of the bottom textured region are varied around their initial values. In particular, in Figures 23 and 24, minimum ﬁlm thickness and power loss are presented against CA, for diﬀerent values if TBs and TBe, respectively. Similarly to the case of hydrophobic sliders, the results of Figure 23 suggest that variation of the starting location of the bottom textured region aﬀects ring performance only during the downstroke period. The largest textured area (0–30%) has the best performance, since it further increases minimum ﬁlm thickness and decreases the corresponding power loss. From Figure 24, it can be concluded that textured areas extending beyond 30% of the piston ring length have an insigniﬁcant eﬀect on the performance parameters. Similarly, smaller textured areas also reduce minimum ﬁlm thickness and increase power loss. Merging the observations of Figures 23 and 24, the optimum design of the bottom textured area is the one that extends from the start of the piston ring face Koukoulopoulos and Papadopoulos 17 Figure 22. One hundered percent engine load, 105 r/min: (a) Minimum film thickness, and (b) power loss against crank angle. Comparison between the conventional (untextured) and the textured ring design. Figure 23. One hundred percent engine load, 105 r/min (a) Minimum film thickness, and (b) power loss against crank angle, for different values of the start of bottom textured region, TBs. Figure 24. One hundred percent engine load, 105 r/min (a) Minimum film thickness, and (b) power loss against crank angle, for different values of the end of bottom textured region, TBe. 18 Proc IMechE Part J: J Engineering Tribology 0(0) Figure 25. (a) Minimum film thickness, (b) power loss against crank angle, for different values of the start of top textured region, TS s. Figure 26. (a) Minimum film thickness, (b) power loss against crank angle, for different values of the end of top textured region, TTe. proﬁle (TBs ¼ 0) until 30% of its total length (TBe ¼ 0.3). increases the minimum ﬁlm thickness, decreases power loss, while aﬀecting insigniﬁcantly the maximum pressure and the cavitation area. Top texturing region In this section, the start and end locations of the bottom texturing region are varied around their initial values. In Figure 25, the starting point (TTs) takes values between 0.6 and 0.8, while in Figure 26, the end point (TTe) takes values between 0.8 and 1. Lower values TTs (60–90% and 65–90%), which are equivalent to larger textured areas, provide a small increase of the minimum ﬁlm thickness and a small decrease of power loss in the upstroke motion, while they decrease the former and increase the latter for the downstroke motion. Variation of TTe aﬀects the ring operation only during the downstroke. According to Figure 26, the largest textured area (70–100%) has the best performance, since it further Texture depth From the above analysis, the best piston ring design with surface texturing is the one with textured areas expanding (a) from 0% to 30%, which improves the operation of the downstroke motion, and (b) from 70% to 100%, which improves the operation of the upstroke motion. These results were obtained for a texture depth equal to 5 mm; in the present section, texture depth values of 2, 8, 10, and 15 mm are also considered. In Figure 27, the eﬀect of diﬀerent texture depth values on the operation of the piston ring is presented. From Figure 27 it can be noticed that increasing dimple depth beyond 5 mm does not improve Koukoulopoulos and Papadopoulos 19 Figure 27. (a) Minimum film thickness, (b) power loss against crank angle, for different values of dimple depth. Here, the optimal textured ring is considered. Figure 28. (a) Minimum film thickness, (b) power loss against crank angle, comparison between the reference case (plain) piston ring and the optimal textured design. Figure 29. Twenty-five percent load, 66.1 r/min (a) Minimum film thickness, (b) power loss against crank angle, comparison between the reference case (plain) piston ring and the optimal textured design (resulted from simulations at 100% engine load). 20 minimum ﬁlm thickness (hmin is actually decreased for a dimple depth value of 15 mm). However, power loss maintains a decreasing trend with increasing dimple depth, achieving a decrease of approximately 30% for dd ¼ 15 mm. Optimal piston ring design with texturing From the parametric analysis of previous sections, the resultant optimal design is characterized by artiﬁcial surface texturing from 0% to 30% and 70% to 100% of the total piston ring surface, with a dimple depth value equal to 10 mm. In Figure 28, a comparison is made between the results of the reference (untextured) and of the optimal textured case. Observing Figure 28, the optimally textured ring design, increases the mean value of minimum ﬁlm thickness over a full cycle by 11%, while mean power loss, over a full cycle, is decreased by 25%. Similarly to the case of hydrophobicity, in order to examine if the above design remains beneﬁcial at lower loads and values of rotational speed, Figure 29 is generated, presenting the behavior of the optimal hydrophobic piston ring design, which resulted from the 100% engine MCR and 105 r/min rotational speed (nominal operating condition of the engine), operating now at 25% engine MCR and 66.1 r/min. Based on the results of Figure 29, it can be observed that a limited performance improvement is also present at lower loads/operating speeds of the engine. Conclusion Approximately 80% of the annual operating cost of modern large cargo vessels corresponds to fuel consumed for vessel propulsion and auxiliary system operation. Engine mechanical losses correspond to approximately 4–6% of the engine Brake horse power (BHP). A quarter of those losses is due to piston ring operation.2 On the other hand, piston ring overhauling/replacement due to wear is usually required after 10,000–30,000 h of operation, depending on ring type, operating conditions, and fuel type.26 Therefore, reducing the friction power losses and increasing operating minimum ﬁlm thickness (thus decreasing wear rate) of the piston ring pack of a Diesel engine, can lead to a substantial reduction of the annual operational cost for a ship, in terms of fuel consumption and maintenance/replacement costs. The results of the present paper demonstrate that piston rings with either slip properties at certain parts of the interacting face and piston rings or peripheral texturing (grooves) exhibit enhanced tribological behavior. In particular, a substantial reduction of power loss, of the order of 56%, is demonstrated when hydrophobicity is applied properly to the ring surface, and of the order of 25%, when parts of the ring surface are properly textured. At the same time, the mean value of minimum ﬁlm thickness is increased Proc IMechE Part J: J Engineering Tribology 0(0) by approximately 53% and 11%, respectively, corresponding to a substantial increase of load-carrying capacity, and to a substantial decrease of the probability of metal-to-metal contact, which is the main cause for ring face wear and degradation. It has been observed that, for both surface treatment technologies, surface treatment is most eﬀective when applied to areas in the ﬁrst and last one third of the piston ring face proﬁle. It is pinpointed that the achieved decrease in friction loss and the simultaneous increase in load capacity result in decreased values of the friction coeﬃcient of the system. The presented results refer to a large two-stroke marine diesel engine operating at its MCR and at maximum shaft rotational speed. Additional simulations demonstrate that improved performance is also attained at lower loads and values of rotational speed, however, the improvement is less pronounced in comparison with that corresponding to nominal engine operation. Declaration of Conflicting Interests The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) received no ﬁnancial support for the research, authorship, and/or publication of this article. References 1. Yukio H. Hydrodynamic lubrication. Tokyo, Japan: Springer, 2005. 2. Clausen NB. Marine diesel engines, how efficient can a two-stroke engine be? Copenhangen, Denmark: MAN Diesel A/S, 2013. 3. Stachowiak G and Batchelor A. Engineering tribology. United Kingdom: Butterworth Heinemann, 2011. 4. Jeng Y-R. Theoretical analysis of piston-ring lubrication part-I fooly flooded lubrication. Tribol Trans 1992; 35: 696–706. 5. Jeng Y-R. Theoretical analysis of piston-ring lubrication part-II staeved lubrication and its application to a complete ring pack. STLE Tribol Trans 1992; 35: 707–714. 6. Wakuri Y, et al. Piston ring friction in internal combustion engines. Tribol Int 1992; 25: 299–308. 7. Livanos GA and Kyrtatos NP. Friction model of a marine diesel engine piston assembly. Tribol Int 2007; 40: 1441–1453. 8. Giacopini M, et al. A mass-conserving complementarity formulation to study lubricant films in the presence of cavitation. ASME J Tribol 2010; 132: 1–12. 9. Elrod HG and Adams ML. A computer program for cavitation and starvation problems. Leeds-Lyon Conference on Cavitation, Leeds University, England, 1974. 10. Elrod HG. A cavitation algorithm. ASME J Lubricat Technol 1981; 103: 350–354. 11. Vijayaraghavan D and Keith TG Jr. Development and evaluation of a cavitation algorithm. STLE Tribol Trans 1989; 32: 225–233. Koukoulopoulos and Papadopoulos 12. Khonsari MM and Fesanghary M. A modification of the switch function in the Elrod cavitation algorithm. ASME J Tribol 2011; 133: 1–4. 13. Fatu A, et al. Wall slip effects in (elasto) hydrodynamic journal bearings. Tribol Int 2011; 44(7–8) DOI: 10.1016/ j.triboint.2011.03.003. 14. Pavlioglou SK, et al. Tribological optimization of thrust bearings operated with lubricants of spatially varying viscosity. ASME GT2014-25292 2014; 137: 1– 10. 15. Guo-Jun M, Cheng WW and Ping Z. Hydrodynamics of slip wedge and optimization of surface slip property. Sci China-Phys Mech Astron 2007; 50: 321–330. 16. Etsion I, Kligerman Y and Shinkarenko A. Improved tribological performance of piston rings by partial surface texturing. Trans ASME 2005; 127: 632–638. 17. Papadopoulos CI, Efstathiou EE, Nikolakopoulos PG, et al. Geometry optimization of textured three-dimensional micro-thrust bearings. ASME J Tribol 2011; 133: 1–14. 18. Xiong S and Wang QJ. Steady-state hydrodynamic lubrication modeled with the payvar-salant mass conservation model. ASME J Tribol 2012; 134: 1–16. 19. Taylor RI. Squeeze film lubrication in piston rings and reciprocating contacts. J Eng Tribol 2015; 229: 1–12. 20. Rothstein JP. Slip on superhydrophobic surfaces. Annu Rev Fluid Mech 2010; 42: 89–109. 21. Neto C, Evans DR, Bonaccurso E, et al. Boundary slip in Newtonian liquids: a review of experimental studies. Rep Prog Phys 2005; 68: 2859–2897. 22. Tsai P, Peters AM, Pirat C, et al. Quantifying effective slip length over micropatterned hydrophobic surfaces. Phys Fluids 2009; 21: 112002. 23. Verho T, Bower C, Andrew P, et al. Mechanically durable superhydrophobic surfaces. Adv Mater 2011; 23: 673–678. 24. Etsion I. State of the art in laser surface texturing. In: ASME, 7th biennial conference on engineering systems design and analysis, J. Tribol 2005; 127(1), 248–253. 25. Kligerman Y, Etsion I and Shinakerenko A. Improving tribological performance of piston rings by partial surface texturing. J Tribol 2005; 127: 632–638. 26. MAN DIESEL & TURBO, Service experience, twostroke engines, 2012. http://marengine.com/ufiles/ MAN-Service_Experience_2012.pdf (accessed 13 October 2017). 21 dd ld Nd E F Fext g h hmax hmin l ls Ls m N n o p p1-2 Pbk pc pch Pel PL pmax R r S SBe SBs STe STs t T Appendix Notation a b B b* bs c CA dcl S slip proportionality factor [m/Pa.s]: ¼ bs/ ring width [m] bore diameter [m] nondimensional slip length: b* ¼ bs/r slip length [m] crown height [m] crank angle [degrees] texture cell length [m] texture density TBe TBs TTe TTs tp dimple depth [m] dimple length [m] number of dimples Young’s modulus of elasticity [Pa] friction force [N] sum of external forces acting on the ring back face [N] cavitation factor film thickness [m] maximum film thickness [m] minimum film thickness [m] ring length along the peripheral [m] nondimensional length of slip area, ls ¼ Ls/b length of slip area [m] grid points along the y direction rotational speed [r/min] grid points along the x direction offset [m] pressure [Pa] pressure between the first and second compression rings [Pa] gas force on the piston ring back face [N] cavitation pressure [Pa] pressure in the combustion chamber [Pa] pretension force of piston ring [N] power loss [W] maximum pressure along the piston ring width [Pa] rod length [m] composite roughness of piston ring surface [m] engine stroke [m] bottom hydrophobic region: nondimensional x coordinate of slip end location bottom hydrophobic region: nondimensional x coordinate of slip start location top hydrophobic region: nondimensional x coordinate of slip end location top hydrophobic region: nondimensional x coordinate of slip start location time [s] tangential tension force of piston ring [N] bottom textured region: nondimensional x coordinate of texture end location bottom textured region: nondimensional x coordinate of texture start location top textured region: nondimensional x coordinate of texture end location top textured region: nondimensional x coordinate of texture start location time points 22 U u v W w x y z l c c Proc IMechE Part J: J Engineering Tribology 0(0) piston speed [m/s] fluid velocity in the x direction [m/s] fluid velocity in the y direction [m/s] load-carrying capacity of fluid [N] fluid velocity in the z direction [m/s] direction along the piston ring profile (streamwise direction) direction along the piston ring circumference (spanwise direction) direction along the film thickness (crossflow direction) lubricant bulk modulus [Pa] lubricant dynamic viscosity [Pa.s] lubricating film fraction ring gap in the peripheral direction [m] dry friction coefficient lubricant density [kg/m3] critical shear stress [Pa] shear stress [Pa] Appendix 1—Reynolds equation for hydrophobic surfaces A small volume of ﬂuid from the lubricating ﬁlm is considered, as seen in Fig. 30, and it is assumed that forces are applied only along the x direction (extension to direction y is trivial). Equilibrium of the element dictates that the forces acting on at the left side of the volume must be equal to the forces acting on the right one. @x pdxdz þ x þ dz dxdz @z @p ¼ p þ dx dydz þ x dxdz ) @x @x @p ) dx dydz ¼ dx dydz @x @z Figure 30. Force equilibrium of a finite volume of fluid. ð19Þ Considering dxdydz 6¼ 0, equation (19) can be divided by dxdydz, which yields @x @p ¼ @x @z ð20Þ A similar equation can be derived for the y direction @y @p ¼ @y @z ð21Þ Pressure is assumed constant along the z direction (ﬁlm thickness direction), thus @p ¼0 @z ð22Þ The shear stress of the lubricant can be expressed with the use of dynamic viscosity and the rate of shear along both the x and y directions as follows x ¼ @u @z ð23Þ y ¼ @v @z ð24Þ where, x and y are the shear stresses acting along the x and y direction, respectively, whereas u and v are the corresponding ﬂuid velocities. Substituting equation (23) into equation (20), and equation (24) into equation (21) yields @p @ @u ¼ @x @z @z ð25Þ Koukoulopoulos and Papadopoulos @p @ @v ¼ @y @z @z 23 ð26Þ Substituting equation (37) into equation (36) yields @p @p h2 h ar C,r U al C,l @x @x 2 C1 h ¼4 ¼ al C1 þ ar C1 þ ar Integrating equation (25) yields @p z2 @p z2 C1 C2 þ C1 z þ C2 ¼ u ) u ¼ þ zþ @x 2 @x 2 ð27Þ 2 C1 ¼ @p @p h ar x h þ ar C,r þ @x 2 þ U þ al C,l h þ ðal þ ar Þ Diﬀerentiating equation (27) yields @p @u z þ C1 ¼ @x @z ð38Þ ð28Þ At this point, slip conditions are introduced at both surfaces uz¼0 @u ¼ U al C,l @z z¼0 u uz¼h ¼ ar C,r z z¼h ð29Þ 93 28 @p h2 < @p = 6 ah @x h þ @x 2 þ Uþ 7 6: 7 6 ar C,r C,l C,l h ;7 7 C2 ¼ U al 6 6 7 h þ ðal þ ar Þ 6 7 4 5 ð39Þ ð30Þ where C,l and C,r are the critical shear stress values for the liner and the piston ring, respectively (value of shear stress above which slip is initiated). The ﬁrst derivative of velocity at z ¼ 0 and z ¼ h are @u ¼ C1 @z z¼0 C2 can be calculated from equation (37) Substituting constants C1 and C2 into equation (27) yields the velocity along direction x u¼ ð31Þ @p z2 @p h 2ar þ h U z þ @x 2 @x 2 h þ ðal þ ar Þ h þ ðal þ ar Þ ar C,r þ al C,l h þ ar þ þU h þ ðal þ ar Þ h þ ðal þ ar Þ ar C,r C,l C,l h @p h 2al ar 2 þ al h al @x 2 h þ ðal þ ar Þ h þ ðal þ ar Þ ð40Þ Similarly, the ﬁrst derivative of velocity at z ¼ h is @u @p ¼ C1 þ h @z z¼h @x ð32Þ Z Velocities at z ¼ 0 and z ¼ h can be calculated as uz¼0 ð33Þ @p h2 C1 C2 þ hþ @x 2 ð34Þ U al C1 C,l @ðuÞ dzþ @x 0 Z h 0 @ðvÞ dzþ @y Z 0 h @ðwÞ dz þ @z Z 0 h @ dz ¼ 0 @t C2 ¼ ð35Þ h @ðuÞ @ dz ¼ @x @x Z h uðzÞdz uðhÞ 0 @h @x ð42Þ Substituting equation (40) into the ﬁrst part of equation (42) results in @p h3 h2 @p h 2ar þ h @x 6 2 @x 2 h þ ðal þ ar Þ 0 U ar C,r þ al C,l þ þ h þ ðal þ ar Þ h þ ðal þ ar Þ h uðzÞdz ¼ ð36Þ h þ ar @p h 2al ar 2 þ al h h @x 2 h þ ðal þ ar Þ h þ ðal þ ar Þ ar C,r C,l C,l h hal h þ ðal þ ar Þ þ Uh Solving equation (35) for C2 C2 ¼ U al C1 C,l Z Z From equations (30), (32), and (34) @p @p h2 C1 C2 þ hþ ar C1 h C,r ¼ @x @x 2 Using the Leibnitz rule for diﬀerentiation of integrals, the ﬁrst term of equation (41) can be written as 0 Use of equations (29), (31), and (33) results in h ð41Þ C2 ¼ uz¼h ¼ Continuity of mass demands that ð37Þ 24 Proc IMechE Part J: J Engineering Tribology 0(0) h2 @p h2 þ 4ðal þ ar Þh þ 122 al ar 12 @x h þ ðal þ ar Þ Uh h þ 2ar þ 2 h þ ðal þ ar Þ 1 h2 al C,l ar C,r þ 2al ar C,l C,r h þ 2 h þ ðal þ ar Þ Thus, the second term of equation (41) is written in total ¼ Z 0 ð43Þ Substituting equation (40) into the second part of equation (42) results in h @ðvÞ @ @p h2 h2 þ 4hðar þ al Þ þ 12al ar 2 dz ¼ y @y @y 12 h þ ðar þ al Þ @ h ar C,r þ al C,l h þ 2ar al C,r þ C,l @y 2 h þ ðar þ al Þ @p @h h ar h þ 2al ar 2 þ @y @y 2 h þ ðar þ al Þ ar C,r h þ ar al C,r þ C,l @h þ @y h þ ðar þ al Þ ð46Þ @p h2 @p h 2ar þ h h @x 2 @x 2 h þ ðal þ ar Þ U ar C,r þ al C,l þ þ þ h þ ðal þ ar Þ h þ ðal þ ar Þ For the third term of equation (41), the following is valid, as the squeeze motion only occurs uðhÞ ¼ Z 2 h þ ar @p h 2al ar þ al h þ U @x 2 h þ ðal þ ar Þ h þ ðal þ ar Þ ar C,r C,l C,l h ) al h þ ðal þ ar Þ @p h ar h þ 2al ar 2 ar uðhÞ ¼ þ U @x 2 h þ ðal þ ar Þ h þ ðal þ ar Þ ar al C,r C,l þ ar C,r h h þ ðal þ ar Þ ð44Þ Thus, the ﬁrst term of equation (41) is written as Z h 0 @ðuÞ @ h2 p h2 þ 4ðal þ ar Þh þ 122 al ar dz ¼ x @x 12 x h þ ðal þ ar Þ @ U h þ 2ar @ 1 h þ þþ @x 2 @x 2 h þ ðal þ ar Þ h al C,l ar C,r þ 2al ar C,l C,r h h þ ðal þ ar Þ @h @p h ar h þ 2al ar 2 @x @x 2 h þ ðal þ ar Þ @h ar U @x h þ ðal þ ar Þ @h ar al C,r C,l þ ar C,r h þ @x h þ ðal þ ar Þ þ ð45Þ A similar procedure is followed for the derivation of velocity v along the y direction. h 0 @ðwÞ @h dz ¼ w ¼ z @t ð47Þ The ﬁnal term of equation (41) is the accumulation ratio, and is equal to Z h 0 @ @ dz ¼ h @t @t ð48Þ Substituting all the three terms for x, y, and z directions in equation (41) of continuity of ﬂow, the result is the Reynolds equation for slip conditions on the solid boundary. @ h2 p h2 þ 4ðal þ ar Þh þ 122 al ar @x 12 x h þ ðal þ ar Þ 2 2 @ @p h h þ 4hðar þ al Þ þ 12al ar 2 þ @y @y 12 h þ ðar þ al Þ U @ h2 þ 2ar h ar ¼ U 2 @x h þ ðal þ ar Þ h þ ðal þ ar Þ @h h ar h þ 2al ar 2 @h @p @p @h þ þ þ @x 2 h þ ðal þ ar Þ @x @x @y @y 2 h al C,l ar C,r þ 2al ar h C,l C,r @ 1 þ h þ ðal þ ar Þ @x 2 ar al C,r C,l þ ar C,r h @h þ @x h þ ðal þ ar Þ @ h ar C,r þ al C,l h þ 2ar al C,r þ C,l h þ ðar þ al Þ @y 2 ar C,r h þ ar al C,r þ C,l @h @ðhÞ þ þ @y @t h þ ðar þ al Þ ð49Þ

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