Xin Q. Diesel Engine System Design

740 Diesel engine system design

the bearings, the seals, the valvetrain, and the accessories) to include their

basic design parameters as inputs. Fujii et al. (1988) attempted to use an

analytical approach to derive the relationship between FMEP and cylinder

bore, stroke, and engine speed. Although their analysis was for a very high

speed gasoline motorcycle engine, their approach is a generic methodology

deserving attention. The most comprehensive level-1 breakdown overall model

of engine friction was developed by Patton et al. (1989) for gasoline engines,

later upgraded by Sandoval and Heywood (2003) at the Massachusetts Institute

of Technology. This model was used by Shayler et al. (2005) to compare

with the light-duty diesel engine test data. The basic idea in the breakdown

model is to assume the coef cient of friction follows the Stribeck diagram.

Therefore, the coef cient of friction is basically proportional to the Stribeck

duty parameter or its square root in the hydrodynamic lubrication, varying

inversely with engine speed in the mixed lubrication, or being a constant in

the boundary lubrication. The friction force can be obtained by multiplying

the friction coef cient by the load (a normal force). The friction terms for

each component can be derived using the FMEP format shown in equation

10.6. The detailed derivations were given by Heywood (1988), Patton et al.

(1989), and Ferguson and Kirkpatrick (2001). A modi ed model including

the correction factors related to lubricant viscosity, piston ring tension, and

cylinder bore surface roughness was summarized by Sandoval and Heywood

(2003).

The level-1 breakdown overall engine friction model is summarized in

the following form:

v

m

FM

EP

FMEPFM

B

EE

E

v

n

E

m

BB

B

E

C

d

B

d

B

BS

EE

BS

EE

n

C

Nd

E

Nd

E

m

Nd

m

BB

Nd

BB

Ln

BB

Ln

BB

B

E

B

E

=

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

1

2

BS

2

BS

2

0

3

2

+

Sn

SSnS

C

Nd

n

C

EE

Sn

EE

Sn

SSnS

EE

SSnS

EB

Nd

EB

Nd

B

E

Cr

anks

CranksCr

haf

hafthaf

v

+

3

22

Nd

22

Nd

4

È

Î

Í

È

Í

È

Í

Î

Í

Î

Í

˘

˚

˙

˘

˙

˘

˙

˚

˙

˚

˙

m

vvv

n

mp

m

E

rt

sr

v

B

E

B

E

C

F

rt

F

rt

F

rt

F

rt

0

5

00

sr00sr

00

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

00

¯

00

˜

ˆ

˜

ˆ

¯

˜

¯

00

¯

00

˜

00

¯

00

Ê

Ë

00

Ë

00

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

+

,

s

00

s

00

ˆ

˜

ˆ

˜

ˆ

˜

ˆ

¯

˜

¯

˜

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

+

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

1

0

2

6

0

3

2

C

N

B

C

Nd

Ln

B

E

B

E

B

v

n

E

Nd

E

Nd

m

Nd

m

Nd

BB

Nd

BB

Nd

Ln

BB

Ln

B

E

B

E

B

+

m

Sn

SSnS

C

EE

Sn

EE

Sn

SSnS

EE

SSnS

PistonAssembly

v

È

Î

Í

È

Í

È

Í

Î

Í

Î

Í

˘

˚

˙

˘

˙

˘

˙

˚

˙

˚

˙

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

+

7

0

m

ˆ

˜

ˆ

˜

ˆ

˜

ˆ

¯

˜

¯

˜

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

◊

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

n

boos

t

am

bien

t

rt

boos

t

am

p

C

F

rt

F

rt

F

rt

F

rt

p

W

+

8

0

,

bien

bbienb

t

Cv

Ga

sL

oading

mp

◊

È

Î

Í

È

Í

È

Í

Î

Í

Î

Í

˘

˚

˙

˘

˙

˘

˙

˚

˙

˚

˙

-

W

(.

)

13

(.13(.

3

9

Cv

9

Cv

Diesel-Xin-10.indd 740 5/5/11 12:00:46 PM



741Friction and lubrication in diesel engine system design

CC

CC+CC

10

CC

10

CC

11

+

mmm

m

v

n

E

m

B

EE

E

VAL

Nn

E

Nn

E

m

Nn

m

BS

EE

BS

EE

n

C

N

n

0

2

BS

2

BS

12

0

1

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

+

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

+

Sn

SSnS

C

Nn

Sn

C

l

EE

Sn

EE

Sn

SSnS

EE

SSnS

E

Nn

E

Nn

VAL

EE

Sn

EE

Sn

v

n

VAL

ma

x

maxma

+

13

14

0

1

m

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

,

..

,

50

5

15

0

1

Nn

.

Nn

.

50

Nn

50

5

Nn

5

BS

n

C

N

ln

E

Nn

E

Nn

VAL

EE

BS

EE

BS

E

VAL

ln

VAL

ln

ma

ln

ma

ln

x

ln

x

ln

maxma

ln

ma

ln

x

ln

ma

ln

VAL

+

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

Sn

SSnS

CC

N

EE

Sn

EE

Sn

SSnS

EE

SSnS

Valvetrain

È

Î

Í

È

Í

È

Í

Î

Í

Î

Í

Î

Í

Î

Í

Î

Í

Î

˘

˚

˙

˘

˙

˘

˙

˚

˙

˚

˙

˚

˙

˚

˙

˚

˙

˚

CC+CC

+

16

CC

16

CC

17

EE

EEEE

A

ccessories

CN

EE

CN

EE

+

EE

+

EE

È

Î

È

Î

È

˘

˚

˘

˚

˘

18

EE18EE

CN

18

CN

EE

CN

EE18EE

CN

EE

2

10.57

where the C

i

are empirical constants, n is a viscosity friction exponent, and

m is a speed exponent. The n and m may have different values for different

components.

The terms in the  rst bracket of equation 10.57 represent the frictions

due to the crankshaft main bearing seals, the main bearings’ hydrodynamic

lubrication, and the turbulent dissipations in transporting the oil through the

bearings, respectively. The term d

B

is the bearing diameter, L

B

is the bearing

length, and n

B

is the number of bearings. Note that the turbulent dissipation

term may be dropped and included in the oil pump work.

The terms in the second bracket represent the friction due to the

hydrodynamic lubrication of the piston assembly in reciprocating motion,

the piston ring friction without gas pressure loading, and the hydrodynamic

lubrication of the connecting rod bearings, respectively. The term F

rt

is

the piston ring tension, m

v

is the lubricant viscosity, and s

sr

is the surface

roughness. The subscript ‘0’ represents a baseline value which is used to

tune the model.

The terms in the third bracket represent the increase in friction due to gas

pressure loading, where p

boost

is the intake manifold boost pressure and W

is the engine compression ratio. Note that these terms can be improved by

replacing them with the more generic terms of peak cylinder pressure.

The  rst two terms in the fourth bracket represent the frictions due to the

boundary lubrication of the camshaft bearing seals, and the hydrodynamic

lubrication of the camshaft bearings, respectively. The parameter n

VAL

is

the number of valves per cylinder, and l

VAL,max

is the maximum valve lift.

The third and fourth terms in the fourth bracket represent the cam–follower

friction. Note that only one of them is to be used (the third for the  at-

faced follower and the fourth for the roller follower). The last two terms

in the fourth bracket represent the friction due to the motions of valvetrain

oscillating components at the tappet–bore interface in the hydrodynamic

or mixed lubrication and the valve stem–guide interface in the mixed or

boundary lubrication.

Diesel-Xin-10.indd 741 5/5/11 12:00:47 PM



742 Diesel engine system design

The terms in the last bracket represent the friction and parasitic losses of

the engine accessories such as the oil pump, the water pump, the fuel pump,

and the alternator. Detailed information on engine and vehicle accessories can

be found in SAE J1343 (2000), J2743 (2007), J1342 (2007), J1341 (2003),

and J1340 (2003).

Note that the viscosity correction is important because the friction power

at engine start-up may be twice as high as the warmed-up engine friction

power (Sandoval and Heywood, 2003). The viscosity friction exponent n may

vary for different components (Shayler et al., 2005). Usually, n = 0.5. The

speed exponent m on the engine speed N

E

or the mean piston speed v

mp

in

equation 10.57 may use a value lower than 1 (e.g., 0.5–0.6) in order to match

experimental data better at low-temperature and low-speed conditions, as

attempted by Shayler et al. (2005). The value of m depends on the exponent

used in the correlation between the coefcient of friction and a Stribeck duty

parameter. Also note that the effects of component clearances and piston skirt

length in equations 10.15 and 10.57 are buried in the empirical constants. If

the effects of the design change on these parameters need to be evaluated,

a level-2 or level-3 friction model needs to be used.

10.8.2 Level-2 engine friction model

Previous research results show that the instantaneous friction torques of the

piston assembly, the crankshaft, the camshaft, and the fuel pump contain

peak values much greater than their cycle-average values. There are two

motivations to upgrade the level-1 engine friction model to a level-2 model:

(1) to predict the instantaneous friction torque with crank angle resolution,

more physics-based and less empirically, while still remaining real-time

capable; and (2) to cover more engine design and operating parameters as

inputs so that the effect of their design change can be evaluated, for example

piston skirt length and width, piston ring tension, bearing length and diameter,

clearances, valve spring preload and stiffness.

The level-2 friction model plays a very important role in engine dynamics

modeling for instantaneous or transient torque prediction and cold start

simulation. The level-2 model needs to analyze the friction of each component

individually. The approach of the level-2 model is to compute the friction

force by either using a Coulomb’s type friction model, i.e., the product of

friction coefcient and normal force, or using the established analytical

formula for a particular component (e.g., the ‘short-bearing’ theory used by

Taraza et al., 2007). Both approaches need to be on an instantaneous crank

angle basis. The normal force can be calculated by engine cycle simulation

and dynamics models, which are used as engine system design tools. Some

friction coefcients can be determined by the Stribeck-type diagram (e.g.,

piston rings), based on a lubrication parameter (as a preferred approach), or

Diesel-Xin-10.indd 742 5/5/11 12:00:47 PM



743Friction and lubrication in diesel engine system design

a duty parameter (traditional approach), or the lm thickness-to-roughness

ratio.

Unlike the level-3 model, the level-2 model does not require detailed

knowledge of the spatial distributions of the lubricating oil lm thickness

and pressure on the component surface. The major difference between the

level-2 and level-3 models is the approach to handle the coefcient of friction.

The level-2 model has an advantage that some friction coefcients can come

from an experimentally determined Stribeck diagram (e.g., piston rings). In

the level-3 analytical models, the Reynolds equation is solved to derive the

coefcient of friction. Many simplifying assumptions and boundary conditions

have to be made. In fact, the engine piston assembly and bearings operate

under transient thermal and dynamic conditions that are difcult to model

accurately by the Reynolds equation of hydrodynamic lubrication.

Instantaneous engine friction models were developed by a group of

researchers at Wayne State University, mainly for diesel engines, starting with

the Rezeka–Henein model (Rezeka and Henein, 1984), followed by a series

of friction research (Gardner and Henein, 1988; Taraza et al., 1996; Henein

et al., 1997; Stanley et al., 1999; Teodorescu et al., 2003). Their models

were reviewed by Ciulli (1993, Fig. 10.23) and summarized by Taraza et

al. (2000, 2007). The Rezeka–Henein model has been used and modied by

others (e.g., Tuccillo et al., 1993), especially a group of researchers at the

National Technical Univesity of Athens (Kouremenos et al., 2001; Rakopoulos

et al., 2002, 2004a, 2004b, 2007; Rakopoulos and Giakoumis, 2009). The

Rezeka–Henein model looks up the friction coefcient from a Stribeck

Average

Instantaneous

0 180 360 540 720

Crank angle (degree)

Engine speed (r/min)

2020

2010

2000

1990

1980

10.23 Trends of angular speed obtained through simulation program

using formulae for average (Millington and Hartles, 1968) and for

instantaneous (Rezeka and Henein, 1984) evaluation of losses (from

Ciulli, 1993).

Diesel-Xin-10.indd 743 5/5/11 12:00:47 PM



744 Diesel engine system design

diagram to calculate the piston ring friction; uses the short-bearing analytical

solution to calculate the bearing friction; and uses the elastohydrodynamic

lubrication theory to calculate the cam–follower friction. The model does not

handle the piston skirt friction in an effective way because a Coulomb’s type

friction model is used to try to calculate the viscous friction of a dynamically

loaded lubricated component on an instantaneous basis.

Several important models on the instantaneous engine friction (i.e., with

crank angle resolution) have been developed. These are:

∑ Dowson et al. (1996) handled the piston skirt friction modeling effectively

by directly using a simpli ed viscous shear stress calculation and avoiding

using the Stribeck diagram for friction coef cient. Their work is the

most important reference for the level-2 engine friction model.

∑ Nakada et al. (1997) presented a simpli ed and effective model of piston

friction.

∑ Coy (1997) used the Stribeck diagram to look up the friction coef cient

and also developed an engine wear model.

∑ Zweiri et al. (2000) provided detailed derivations for a similar model

to the Rezeka–Henein model.

The level-2 engine friction model is summarized in the following form:



WW

fE

WW

fE

WW

fO

WW

fO

WW

ilRin

gf

WW

gf

WW

Co

WW

Co

WW

mp

WW

mp

WW

ressi

WW

ressi

WW

onRings

WW

onRings

WW

f

WW

f

WW

fE,fE

()

WW()WW

WW =WW

+

WW +WW

fO

+

fO

WW

fO

WW +WW

fO

WW

ilRin

+

ilRin

gf

+

gf

WW+WW

,,

fO,,fO

ilRin,,ilRin

gf,,gf

f

()

f

()

WW()WW

f

WW()WW

,,

,

+

PistonSkir

tf

+

tf

+

Bearings

fC

,fC,

W

tf

W

tf

W

fC

W

fC



am

aama

fA

ccessories

fV

alveO

fValveOfV

scilla

tin

gf

Se

WW

fA

WW

fA

ccessories

WW

ccessories

fV

WW

fV

W

gf

W

gf

++

WW++WW

fA

WW

fA

++

fA

WW

fA

ccessories

WW

ccessories

++

ccessories

WW

ccessories

+

gf

+

gf

,,

fA,,fA

ccessories,,ccessories

fV,,fV

,





al

aala

s

alsal

vf

ri

vfrivf

nm

pO

nmpOnm

ilRin

gc

fr

in

mp

Cf

ca

Cf

ca

vf

Cf

vf

Fv

nm

Fv

nm

Cf

gc

Cf

gc

av

Cf

av

fr

Cf

fr

Fv

in

Fv

in

=

[

]+

pO

]+

pO

ilRin

]+

ilRin

gc

]+

gc

[

gc

]

SS

vf

SS

vf

ri

SS

ri

vfrivf

SS

vfrivf

nm

SS

nm

pO

SS

pO

nmpOnm

SS

nmpOnm

gc

SS

gc

CfSSCf

ca

Cf

ca

SS

ca

Cf

ca

vf

Cf

vf

SS

vf

Cf

vf

FvSSFv

nm

Fv

nm

SS

nm

Fv

nm

]+SS]+

pO

]+

pO

SS

pO

]+

pO

ilRin

]+

ilRin

SS

ilRin

]+

ilRin

gc

]+

gc

SS

gc

]+

gc

[SS [

gc

[

gc

SS

gc

[

gc

Co

CCoC

mp

ressi

onRings

ca

v

vm

p

vmpvm

o

L

c

C

v

vm

v

vm

h

A

P

+

d

AdA

m

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

È

Î

Í

È

Í

È

Î

Í

Î

Ú

0

Ú

0

Ú

˘˘˘

˚

˙

˘˘˘

˙

˘˘˘

˚

˙

˚

PistonSkir

t

bear

ings

BB

Bearings

Fr

bear

Fr

bear

ings

Fr

ings

BB

Fr

BB

N

BB

N

BB

f

+ [

]

+

[

S

fr

ffrf

f

fr

f

in

Ca

m

pu

mp

A

ccessori

e

Fv

in

Fv

in

mp

]+

Ca

]+

Ca

m

]+

m

+

S



h

◊◊◊

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

È

Î

Í

È

Í

È

Î

Í

Î

˘

˚

˙

˘

˙

˘

˚

˙

˚

sss

fV

alveO

fValveOfV

scilla

tin

gf

Seal

s

SealsSeal

WW

fV

WW

fV

alveO

WW

alveO

fValveOfV

WW

fValveOfV

scilla

WW

scilla

tin

WW

tin

gf

WW

gf

+

WW+WW

gf

WW

gf

+

gf

WW

gf

,,

fV,,fV

alveO,,alveO

fValveOfV,,fValveOfV

scilla,,scilla

tin,,tin

gf,,gf



10.58

where the friction coef cient f

fri

, normal load F

n

and relative velocity v are

different for each component, and



m

is the pump  ow rate, h

pump

is the

pump ef ciency, C

cav

is a cavitation factor accounting for the impact of

cavitated or ruptured oil  lm on friction reduction. It should be noted that if

the friction coef cient already includes the effect of cavitation, C

cav

should

be set to 1 to avoid double counting. C

cav

re ects the effective area of the

oil  lm contributed by the lubricant for viscous shear friction, as opposed

to the air, the gas, or the vapor, in the cavitated region. C

cav

is dependent

on the severity of the cavitation and the degree of  lm rupture and air/

Diesel-Xin-10.indd 744 5/5/11 12:00:48 PM

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745Friction and lubrication in diesel engine system design

vapor lling in the gaps. Therefore, C

cav

can be a complex function of the

component’s dynamic eccentricity (for piston skirt and bearings), dynamic

tilting (for skirt), lateral squeeze velocity, tilting velocity (for skirt), and

sliding speed, etc. Each term in equation 10.58 needs to be calculated on

the instantaneous basis if possible, and its detailed formula can be found in

the above mentioned references.

The piston ring friction can be calculated using one of the following

methods: (1) using a pre-established Stribeck-type diagram to look up the

friction coefcient by using a duty parameter or lubrication parameter that

is a function of lubricant viscosity, velocity, load and contact geometry; and

(2) solving the one-dimensional Reynolds equation analytically.

The piston skirt and bearing frictions should not be calculated using a

Coulomb’s type friction model together with a friction coefcient looked

up from a Stribeck diagram because the piston skirt and engine bearings are

dynamically loaded components, and therefore the zero normal load occurs

with high viscous shear friction force at some crank angle locations within

an engine cycle. Another reason for not using a Coulomb’s type friction

model for the piston skirt and the bearings is that the Coulomb’s type model

would bias the loaded side to over-predict the overall effect on friction when

the assembly weight or the cylinder pressure increases. The viscous shear

friction forces acting on the piston skirt and the bearings should be calculated

using the viscous friction model shown in equation 10.58 (an example for

the piston skirt).

The piston pin friction force can be calculated by using a boundary friction

coefcient 0.08–0.12 multiplied by the normal acting load. The friction

forces of the connecting rod big-end bearing, the crankshaft main bearing,

and the camshaft bearings can be calculated by using the models described

in Section 10.6. The connecting rod big-end bearing often operates in the

mixed lubrication regime with high wear due to its relatively slow journal

velocity. Sometimes its friction force can be approximated using the same

approach as the small-end bearing (piston pin) for simplicity. The total normal

force acting on a crankshaft main bearing can be simplied as a summation

of the reactions produced by the two adjacent cranks/cylinders (Taraza et

al., 2000). Other related information on journal bearing calculations can be

found in Schnurbein (1970), Kozhevnikov (1974), and Mourelatos et al.

(1987). Taraza et al. (2000, 2007) provided detailed models for the friction

calculations of the bearings, the cam, and the engine accessories.

The friction torque during engine transients is believed to be different

from its steady-state value at the same engine speed and fueling conditions

(Winterbone and Tennant, 1981). Rakopoulos et al. (2004b, 2007) provide

detailed discussions on this topic.

Diesel-Xin-10.indd 745 5/5/11 12:00:48 PM

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746 Diesel engine system design

10.8.3 Level-3 engine friction model

The level-3 engine friction model refers to solving the Reynolds lubrication

equation at different levels of complexity, from the simplest rigid-body fully-

ooded hydrodynamic lubrication without surface roughness to the most

complex thermo-elastohydrodynamic lubrication including the effects of

surface roughness and advanced algorithms of lubricant cavitation. Usually,

the level-3 model is not real-time capable if a two-dimensional Reynolds

equation needs to be solved by numerical computation at each time step in

the coupled lubrication–dynamics simulation. The motivations to research

the level-3 model include the following:

1. Solve the one-dimensional Reynolds equation for the piston ring to

deduce the friction coefcient to facilitate the piston ring modeling in

the level-2 engine friction model. Note that it is possible to solve the

one-dimensional Reynolds equation analytically with fast computing

speed.

2. Understand the potential of component design changes in friction reduction

with acceptable sufcient oil lm thickness.

3. A large amount of non-real-time engine friction data can be produced by

using the level-3 model and then converted to a real-time friction model

by using surface t or neural network techniques for engine controls

development and transient system design. The work in this area was

introduced by Wilhelm et al. (2007) and Meng et al. (2007a).

Detailed modeling information on the level-3 models is elaborated in the

earlier sections. It is worth noting that the work conducted by Goenka et al.

(1992) and Paranjpe and Cusenza (1992) at General Motors provides good

examples of the level-3 modeling. Livanos and Kyrtatos (2006) summarized

the solutions of the Reynolds equations for the piston rings, the piston skirt,

and the bearings.

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Xin Q. Diesel Engine System Design

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