close

Вход

Забыли?

вход по аккаунту

?

1350650117736638

код для вставкиСкачать
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: lefteris.koukoulopoulos@gmail.com
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
Документ
Категория
Без категории
Просмотров
2
Размер файла
2 472 Кб
Теги
1350650117736638
1/--страниц
Пожаловаться на содержимое документа