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Modeling of Oil Film Thickness in Piston
Ring/Liner Interface
Cristiana Delprete and Abbas Razavykia Politecnico di Torino, Department of Mechanical and Aerospace Engineering, Torino, Italy
Email: {cristiana.delprete, abbas.razavykia}@polito.it
Abstract—The correct understanding of piston ring/liner
lubrication condition has a primary importance in order to
improve internal combustion engine efficiency in terms of
oil consumption and friction losses. Analytical and
numerical investigation of piston ring-pack becomes then a
reliable tool for evaluating piston ring/liner interface
lubrication mechanism. Main aim of this paper is to
examine the effects of technical aspects, such as ring
geometry and operating condition on modeling of piston
ring/liner lubrication. An analytical model based on
lubrication theory under hydrodynamic regime is here
presented and discussed. The model can represent a useful
tool for designing low friction engine components and it can
be applied to develop reliable friction models to predict
actual engine output.
Index Terms—engine, piston ring, modeling, lubrication
I. INTRODUCTION
Energy costs and increasing environmental concerns
lead to develop more efficient internal combustion
engines (ICE). One technique to improve engine
efficiency and reduce oil consumption and emission is to
reduce the friction losses on lubricated surfaces of the
engine [1], [2].
It is widely recognized that the piston ring-pack is the
major contributor to the power losses in reciprocating
engines therefore there is a pressing need to have insight
into lubrication mechanism of piston ring-pack [3], [4].
Decreasing piston assembly friction is an important way
to improve engine efficiency in terms of oil consumption,
emission and fuel consumption. The performance of
piston rings in ICE widely received the researcher’s
attention. Piston rings act as sealing between the liner and
the piston and can be considered as slider bearings;
relevant literatures [5], [6] report that the piston ring
assembly accounts for 20% to 30% of the total frictional
losses, making it imperative for the piston ring
tribological performance to be understood thoroughly.
Today, the automotive industries are under great
pressure to reduce emissions and increase fuel efficiency;
therefore, it is necessary to make a great effort to realize
how to design piston rings with better tribological
performance. Many factors are related to the tribological
behavior of piston ring/liner interface, e.g. the shape of
ring face profile, the ring width, the elastic characteristics
Manuscript received January 1, 2017; revised April 11, 2017.
of the ring, the surface topography and the operating
condition [7]. Due to some problems associated with
experimental observation, such as high cost of facilities
and being time consuming, mathematical modeling
becomes reliable tool to study about engine tribological
performance. The main aim of mathematical modeling is
to describe the different aspects of the real world
problems, their interaction, and their dynamics through
mathematics. Analytical and numerical investigation of
piston ring-pack recently received the attention of
researchers to assess the ring/liner interface lubrication
mechanism. The modeling of piston rings lubrication
characteristics, has primary importance due to two main
reasons. The first is to develop analytical tools to assess
and understand the contribution of each part in frictional
losses in order to direct designers to improve ICE
efficiency; the second is to develop reliable engine
friction models that can be applied in transient engine
simulation to predict the actual engine output.
In the present paper an analytical model of the piston
ring/liner lubrication is presented. Since, during engine
cycle, piston rings are subjected to hydrodynamic, mixed
and boundary lubrication conditions, but they mainly
enjoy hydrodynamic lubrication regime, therefore, the
model is based on lubrication theory under hydrodynamic
regime and takes into the account, the geometry of the
ring and operating condition, in the mathematical
description of the piston ring/liner lubrication. The
presented study is a part of an on-going project that
provides a detailed model, in comparison with the two
most accepted models [8], [9], and clearly describes the
procedure to evaluate the piston ring lubrication.
II. THEORETICAL MODEL
The following assumptions were made during the
modeling:
Ring is fully engulfed and there is no cavity within the oil film thickness;
Oil film thickness is circumferentially uniform;
Lubricant is Newtonian and incompressible;
Thermal and elastic deformation of ring and liner are neglected;
Oil viscosity and density are constant. The piston ring is treated as dynamically loaded
reciprocating bearing, considering sliding and squeeze
action. Reynolds equation has been used as governing
© 2017 Int. J. Mech. Eng. Rob. Res.
International Journal of Mechanical Engineering and Robotics Research Vol. 6, No. 3, May 2017
doi: 10.18178/ijmerr.6.3.210-214210
-
equation to estimate the generated hydrodynamic
pressure in piston ring/liner interface [10]. The main
purpose of a compression ring is to act as a gas seal for
the combustion chamber and prevent leakage. Piston ring
undergoes pressure-loading variation throughout the
engine cycle. The rings are manufactured with a small
elastic force to push the ring against the liner; the gas
pressure acting on the inner face of the ring substantially
enhances the ring elastic force. It is assumed that a thin
oil film separates the compression rings from the liner
and thus Reynolds equation can be used to determine the
film thickness throughout the engine cycle [8].
It is well known that lubrication mechanism of loaded
rolling/sliding bodies can be classified into three
categories: hydrodynamic, boundary and mixed
lubrication. In the full film lubrication (hydrodynamic
lubrication), the lubricant film is sufficient thick to
sustain the load and asperity contact is negligible.
Boundary lubrication deals with the condition that
lubricant thickness is thin and the load is supported
mainly or completely with asperity contacts. Mixed
lubrication is the transition region between the two
previous mentioned lubrication regimes and refers to a
condition in which the load is sustained with lubricant
film and asperity contacts. Mixed lubrication occurs
when the load is high, speed or viscosity is low, due to
high temperature [11].
For the piston ring lubrication analysis and to solve the
Reynolds equation, it is necessary to determine the ring
face shape, the piston ring sliding speed, the cyclic
variation of piston ring loading and the oil viscosity. In
the present model the ring was considered stationary and
the liner is sliding in opposite direction to determine the
coordinate points of lubricant and ring face contact on
ring face axially. Considering an axisymmetric condition
between piston and liner, the one-dimensional Reynolds
equation can be applied to examine the piston ring/liner
interface lubrication:
(1)
where h is nominal oil film thickness (m), x the spatial
coordinate (m) along the cylinder axis, t time (s), p mean
hydrodynamic pressure (Pa), η oil viscosity (Pa·s), and U
instantaneous piston velocity (m/s).
During the engine operating cycle, ring lift may occur
in the piston groove to satisfy the force balance. Due to
negligible axial movement of the ring in the groove, the
same speed of the piston was considered for the ring.
Referring to a centered crank mechanism layout, the ring
velocity is:
(2)
where R is crank radius (m), ω crankshaft angular
velocity (rad/s), θ the crank angle (rad), and L the
connecting rod length (m).
It is recognized that any shape of the ring face after
running-in time undergoes wear and becomes parabolic
[9], as shown in Fig. 1. A generic parabolic ring face
profile can be expressed as:
(3)
where c is the ring crown height (m), b the ring width (m),
and o the ring face offset from the center of the ring (m).
Figure 1. Schematic representation of piston ring/liner interface and boundary pressure distribution.
If the ring face offset is toward the combustion
chamber, o is positive, otherwise it is negative.
As the minimum oil film thickness is a function of
time, hmin
= hmin
(t) , the variation of the nominal oil film
thickness with respect to time and ring face profile can be
expressed as a function of position (x) and time (t):
(4)
Considering a null ring face offset (o= 0), (4) becomes:
h(x,t) = hmin
(t)+4c
b2x2 = h
min+Bx2 (5)
where B is the curvature of the ring profile (m-1).
Substituting (5) in (1) and integrating two times with
respect to x, the hydrodynamic pressure is:
© 2017 Int. J. Mech. Eng. Rob. Res.
International Journal of Mechanical Engineering and Robotics Research Vol. 6, No. 3, May 2017
¶¶x
h3¶p¶x
æ
èç
ö
ø÷ = -6hU
¶h¶x
+12h¶h¶t
U » wR sinq+ R2L
sin 2qæ
èç
ö
ø÷
p =-6hUI0 (x)+12hw¶h¶q
I1(x)+CI2 (x)+ D (6)
where I0(-x) = -I0(x) , I1(-x) = I1(x) , I2 (-x) = -I2(x) and:
I0(x) =1
2hmin Bhmintan-1 x
B
hmin
æ
èçç
ö
ø÷÷+
x
2hmin hmin +Bx2( )
I1(x) =-1
4B hmin + Bx2( )2
I2 (x) =3
8hmin2 Bhmin
tan-1 xB
hmin
æ
èçç
ö
ø÷÷+
x
4hmin hmin + Bx2( )2
+
+ 3x
8hmin2 hmin + Bx
2( )
h = c
b2+ o
æ
èç
ö
ø÷
2(x -o)2
211
h(x,t) = hmin (t)+h = hmin +c
b2+o
æ
èç
ö
ø÷
2(x - o)2
-
Three different scenarios can be considered during the
analytical modeling of lubrication mechanism between
ring face and liner; these scenarios are classified on the
basis of the considered boundary conditions and
assumptions. In the first one (Fig. 2(a), fully flooded
condition) it is assumed that a sufficient oil quantity is
available on the liner and the oil covers the entire ring
face. Moreover, there is no cavitation and oil film rupture
within lubricant film. In the second one (Fig. 2(b),
starvation condition) the oil partially covers the ring face
and some areas of the ring face are exposed to the gas. In
this case, the load imposed by the gas behind the ring and
the ring tension (stiffness) are sustained by the generated
hydrodynamic pressure and the boundary gas pressures
acting on the uncovered part of the ring face. In the last
one (Fig. 2(c), cavitation condition) cavitation in a fluid
that is recognized as the formation of dissolved gas
bubbles within lubricant film due to that oil cannot
sustain large and continuous negative pressure. This
situation is often took place if the mechanical
components in relative motion, are separated by a
lubricant film, such as journal bearings and piston ring-
liner conjunction. In the piston ring-liner interface,
cavitation is caused by sudden reduction in lubricant
pressure at the diverging part of the ring face that results
in transition of oil from liquid form to gas-liquid mixture
[12], [13].
Different boundary conditions can be applied to
consider oil starvation (when oil is not sufficient to cover
the ring face), gas cavitation (when dissolved gas cavity
or cavities appear within the oil film), and fully flooded
condition (when oil covers completely the ring face). The
simplest solution to determine the integration constants C
and D of (6) is obtained excluding the cavitation
condition and assuming that there is no oil film rupture
(therefore also the starvation condition is excluded). The
only boundary conditions are then the gas pressure at
inlet and outlet, so-called fully flooded condition is
represented in Fig. 3.
Therefore for the first compression ring in upward
stroke, the pressure acting on leading edge is the
combustion chamber pressure and the pressure on trailing
edge is the gas pressure between first and second
compression rings. Fig. 4 shows the hydrodynamic
pressure distribution on the face of the piston ring,
considering fully flooded boundary conditions during
downward and upward stroke.
Figure 2. Piston ring/liner interface scenarios: (a) fully flooded condition, (b) starvation condition, (c) cavitation condition.
Figure 3. Fully flooded boundary condition.
Figure 4. Hydrodynamic pressure distribution on ring face.
Fully flooded boundary conditions are expressed as:
© 2017 Int. J. Mech. Eng. Rob. Res.
International Journal of Mechanical Engineering and Robotics Research Vol. 6, No. 3, May 2017
Cup =1
a +b+ gp2 - p1+6hw
a
2hm in hm in+ Ba2( )
é
ë
êêê
ì
íï
îï
+
+ 1
2hm in hminBtan-1 a
B
hm in
æ
èçç
ö
ø÷÷
ù
ûúú+
-12hw dhdq
1
4B hm in+ Ba2( )2
- 1
4hm in2 B
é
ë
êêê
ù
û
úúú
ü
ýïï
þïï
(7)
Dup = p2 +3hw
Bh2dh
dq (8)
Cdown =1
a +b+ gp2 - p1+6hw
a
2hm in hm in+ Ba2( )
é
ë
êêê
ì
íï
îï
+
+ 1
2hm in hminBtan-1 a
B
hm in
æ
èçç
ö
ø÷÷
ù
ûúú+
+12hw dhdq
1
4B hm in+ Ba2( )2
- 1
4hm in2 B
é
ë
êêê
ù
û
úúú
ü
ýïï
þïï
(9)
212
p = p2 @ x = b 2 = a and p = p1 @ x = 0 , during downward stroke of the piston;
p = p1 @ x = -b 2 = -a and p = p2 @ x = 0 , during upward stroke of the piston. Substituting the boundary conditions in (6), the integration constants C and D for upward and downward stroke can be written as reported from (7) to (10).
-
© 2017 Int. J. Mech. Eng. Rob. Res.
International Journal of Mechanical Engineering and Robotics Research Vol. 6, No. 3, May 2017
Ddown = p1+3hwBh2
dh
dq (10)
with
a =a
4hmin hmin + Ba
2( )2, b =
3a
8hmin2 hmin + Ba
2( ),
g =3
8hmin2 Bhmin
tan-1 aB
hmin
æ
èçç
ö
ø÷÷ .
Referring to Fig. 3, the resultant force Fring,gas
contributed by the ring elasticity and the gas pressure behind the ring is:
Fring,gas(t) = b pring + pgas( ) = b 2TbD + pgasæ
èç
ö
ø÷ (11)
where pring is the ring elastic pressure (Pa), pgas the gas
pressure (Pa) behind the ring (note that pgas = p1 if p1 > p2 ,
and pgas = p2 if p1 < p2 ), T the ring tangential force (N),
and D the cylinder bore diameter (m). Foil , hydrodynamic force per unit length (N/m)
exerted by the oil on the ring face, is different in upward and downward stroke due to cyclic variation of loads and pressure:
Foil,up (t) = poil dx-a
0
ò + p2a (12)
Foil,down (t) = poil dx0
a
ò + p1a (13)
Since the problem has to be solved in quasi-steady-state condition, Fring,gas must be equal to Foil at any time
instant, i.e. at any crank angle. The integration of (12) and (13) can be analytically
solved obtaining:
Foil = -6hwJ0(x)+12hwdh
dqJ1(x)+CJ2(x)+ Dx (14)
where J0(-x) = J0(x) , J1(-x) = -J1(x) , J2(-x) = J2(x) and:
J0(x) =x
2hmin Bhmintan-1 x
B
hmin
æ
èçç
ö
ø÷÷
J1(x) = -1
8Bhmin Bhmintan-1 x
B
hmin
æ
èçç
ö
ø÷÷-
x
8Bhmin hmin + Bx2( )
J2 (x) =3x
8hmin2 Bhmin
tan-1 xB
hmin
æ
èçç
ö
ø÷÷-
1
8Bhmin hmin + Bx2( )
Combining (12) and (13) with (14) it follows:
Foil,up (t) = 6hwJ0(a)-12hwdh
dqJ1(a)+
+Cup J2 (0)- J 2(a)éë ùû+ Dupa+ p2a (15)
Foil,down (t) = -6hwJ0 (a)-12hwdh
dqJ1(a)+
+Cdown J2 (a)- J2 (0)éë ùû+ Ddowna+ p1a
(16)
III. SOLUTION METHODOLOGY
The well-established method applied to calculate the cyclic variation of oil film thickness in piston ring/liner interface is to consider the radial velocity of the ring dh dq and the minimum oil film thickness hmin , at
selected increments of the crank angle such that loads acting on piston ring and reaction force experience radial equilibrium throughout the engine cycle. This encourages the march out a solution from any assumed starting condition [14].
Combining (15) with (11) and (16) with (11) in upward and downward stroke respectively, both dh dq and hmin
are unknowns. Oil film thickness variation with respect to crank angle variation or radial velocity of the ring dh dq can be calculated if an estimation of hmin is
available with respect to a some crank angle, at which the oil film thickness can be expected to change only slightly. Starting at mid-stroke position, assumed qi-1 as value of
current crank angle, and neglecting dh dq , the hmin estimation can be made; at subsequent crank
angle qi , this hmin estimation is used to calculate the value
of dh dq . Now, by knowing hmin at previous crank angle
and dh dqat current crank angle qi , hmin can be updated:
hm in, i = hm in, i-1+Dqdh
dq (17)
where Dq is the crank angle increment (rad).
Figure 5. Flow chart to calculate the minimum oil film thickness.
213
-
Based on the calculated minimum oil film thickness
and knowing the roughness of ring and liner surfaces it is
possible to identify the existing lubrication mechanism.
In hydrodynamic lubrication condition, a sufficient
quantity of oil is available to separate the ring face and
the liner surfaces, such that there is no asperity contact
between them. The transition from pure hydrodynamic
lubrication to mixed lubrication occurs [15] when the
following criteria is met:
h
min
Raring
2 + Raliner
2< 4 (18)
where Raring
and Raliner
are respectively the roughness of
the ring face and the liner internal surface (m). Fig. 5
illustrates the numerical procedure to calculate the
minimum oil film thickness.
IV. CONCLUSION
An analytical model of piston ring/liner lubrication
under hydrodynamic condition is presented. The ring is
treated as a dynamically loaded reciprocating bearing,
considering sliding and squeeze actions. Reynolds and
load equilibrium equations are used as governing laws.
The effect of the ring geometry and operating condition
are taken into account. The numerical solution
methodology to solve Reynolds equation and force
equilibrium is also presented.
REFERENCES
[1] S. C. Tung and M. L. McMillan, “Automotive tribology overview of current advances and challenges for the future,” Tribol. Int., vol.
37, pp. 517-536, 2004. [2] D. F. Li, S. M. Rohde, and H. A. Ezzat, “An automotive piston
lubrication model,” ASLE Trans., vol. 25, pp. 151-160, 1983. [3] J. B. Heywood, Internal Combustion Engine Fundamentals,
McGraw-Hill, 1988.
[4] P. Economou, D. Dowson, and A. Baker, “Piston ring lubrication- Part 1: The historical development of piston ring technology,” J.
Lub. Tech., vol. 104, pp. 118-126, 1982. [5] Y. Wakuri, T. Hamatake, M. Soejima, and T. Kitahara, “Piston
ring friction in internal combustion engines,” Tribol. Int., vol. 25,
pp. 299-308, 1992.
[6] C. M. Taylor, Engine Tribology, Elsevier, 1993. [7] E. H. Smith, “Optimising the design of a piston-ring pack using
DoE methods,” Tribol. Int., vol. 44, pp. 29-41, 2011.
[8] L. Ting and J. Mayer, “Piston ring lubrication and cylinder bore wear analysis, Part I – theory,” J. Lub. Tech., vol. 96, pp. 305-313,
1974.
[9] Y. Jeng, “Theoretical analysis of piston-ring lubrication Part-I fully flooded lubrication,” Tribol. Int., vol. 35, pp. 696-706, 1992.
[10] P. Nagar and S. Miers, “Friction between piston and cylinder of an IC engine: A review,” SAE Technical Paper 2011-01-1405.
[11] H. Rahnejat, Tribology and Dynamics of Engine and Powertrain: Fundamentals, Applications and Future Trends, Elsevier, 2010.
[12] M. Priest, D. Dowson, and C. M. Taylor. “Theoretical modelling of cavitation in piston ring lubrication,” J. Mech. Eng. Sci., vol. 214, pp. 435-447, 2000.
[13] W. F. Chong, M. Teodorescu, and N. D. Vaughan, “Cavitation induced starvation for piston-ring/liner tribological conjunction,” Tribol. Int., vol. 44, pp. 483-497, 2011.
[14] D. Dowson, B. L. Ruddy, and P. N. Economou, “The elastohydrodynamic lubrication of piston rings,” in Proc. Royal
Society A: Mathematical, Physical and Engineering Sciences, vol.
386, issue 1791, pp. 409-430, 1983. [15] S. K. Bedajangam and N. P. Jadhav, “Friction losses between
piston ring-liner assembly of internal combustion engine: A review,” Int. J. Sci. and Res. Publ., vol. 3, pp. 1-3, 2013.
Cristiana Delprete received her MSc in Mechanical Engineering from Politecnico di
Torino (Italy) in 1988, and PhD in Applied Mechanics Mechanical Systems and
Structures in 1993.
From 1991 to 1998 she was Assistant Professor and from 1998 to present she is
Associate Professor of Machine Design and Construction of Politecnico di Torino.
She lead the Research Group DePEC (Design
of Powertrain and Engine Components - Materials, Experimental tests, Numerical simulations) of Politecnico di Torino, since 2001. Since 2005
she is member of SAE. Her research activity is focused on: design and analysis of engine components and subsystems, metal replacement in
engine structural design, fatigue and thermo-mechanical fatigue life
estimation, numerical simulation and experimental characterization of materials and components.
Abbas Razavykia received his BSc in the
field of industrial engineering (industrial
technology) in 2009 from QIAU, Iran. He has worked as Manufacturing and Process
Engineer and Project Manager in several companies. After gaining some experience in
industry, he started his post education in the
field of Mechanical Engineering (Advanced Manufacturing Engineering) at UTM, and
received his MSc in 2014. He has started his PhD in 2015 at Politecnico di Torino, Italy.
His interesting research areas are Mechanical Engineering, Production
Planning and Control, Tribology, Optimization and Simulation and Modeling.
© 2017 Int. J. Mech. Eng. Rob. Res.
International Journal of Mechanical Engineering and Robotics Research Vol. 6, No. 3, May 2017
214
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