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

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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