Journal of Materials Processing Technology 140 (2003) 569–575

Modeling of the inlet zone in the mixed lubricationsituation of cold strip rolling

Cheng Lu∗, A. Kiet Tieu, Zhengyi JiangFaculty of Engineering, University of Wollongong, Wollongong, NSW 2522, Australia

Abstract

In this paper a method to simulate the oil thickness and length of elastic deformation in the inlet zone of cold rolling has been developed.The mixed film lubrication model was adopted to describe the behavior of the lubricant and asperity deformation. The elastic Von Karmanequation was used to describe the elastic deformation of strip in the inlet zone. The length and lubricant film thickness of the inlet zone canbe obtained by a numerical method. Results of simulations show that the reduction, rolling speed, back tension have a significant influenceon the lubricant film thickness and the inlet zone length.© 2003 Elsevier B.V. All rights reserved.

Keywords:Mixed film regime; Hydrodynamic inlet; Cold rolling; Elastic deformation; Full film zone

1. Introduction

Lubrication in cold strip rolling is of great importance inreducing rolling force and roll wear and improving the stripsurface quality. In the past decades many attempts to modelthe lubrication of the cold strip rolling have been carriedout. Cheng presented a plastohydrodynamic lubricationmodel based on EHD theory[1]. Bedi and Hiller integratedhydrodynamic lubrication and plasticity, and minimized thedissipated energy in the work zone to calculate the filmthickness[2]. Some studies, such as Wilson and Walowit[3] and Lugt et al.[4,5], focused on the full film regime. Inpractical rolling operation, the mixed film regime prevailsin the roll bite. The mixed film regime of the work zone hasbeen analyzed by a number of researchers including Wilsonand Chang[6], Chang and Wilson[7] and Qiu et al.[8].

In the lubrication of cold strip rolling, the inlet zone playsa dominant role in providing an appropriate boundary condi-tion for the calculation of the plastic work zone. Wilson andWalowit [3] used the full film regime to present a well-knownfilm thickness formula. Saxena et al.[9] also developed asimilar equation. Lugt et al.[4,5] investigated the effect ofthe elastic deformation of strip in the inlet zone based onthe full film regime. Chang and Wilson[7] modeled the in-let zone in the mixed film regime, but the rigid inlet zone(E = 0) was considered in this model. In all the above mod-els the inlet zone length was assumed to be infinite.

∗ Corresponding author. Tel.:+61-2-4221-4549; fax:+61-2-4221-3101.E-mail address:chenglu@uow.edu.au (C. Lu).

This paper developed a mixed film lubrication model inthe inlet zone. Elastic deformation of the strip has beenincorporated in the model. The full film zone before theasperity contact starts in the inlet zone, which determinesthe lubrication effect of the emulsion, has also been inves-tigated. The inlet zone length, the full film zone length andthe lubricant film thickness can be obtained by a numericalmethod. The influence of the elastic deformation of the strip,reduction, rolling and speed, tension are studied. The modelhas been used successfully up to 42 m/s rolling speed.

2. Theoretical model

2.1. General

The schematic of the rolling process is given inFig. 1.The back tensiontb and front tensiontf are applied. Becauseof symmetry, only half the configuration is considered. Theoverall contact area is divided into three zones: the inletzone, the work zone and the outlet zone.

An enlargement at pointxc is drawn at the bottom partof Fig. 1. The strip is considered to be elastic in the inletzone. The inlet zone includes two parts: zone I and zoneII. The boundary between two parts is the point where theasperity contact starts. In zone I there is no asperity contactand the full film regime prevails. Therefore, it is also calledthe full film zone in the remainder of this paper. In zone IIthe asperity will be deformed in carrying part of the load.In the work zone where the plastic deformation of the striptakes place the mixed film lubrication regime prevails. In the

0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0924-0136(03)00794-5

570 C. Lu et al. / Journal of Materials Processing Technology 140 (2003) 569–575

Nomenclature

A fractional contact areac, C constantc0, c1 flow stress constantE non-dimensional strain rateEr, Es Young’s modulus of the

roll and the strip (Pa)h nominal surface separation (mm)ht, Ht average film thickness (mm);

Ht = ht/Rqk shear flow strength of strip

material (Pa)l asperity half-pitch (mm)m adhesion coefficientp, P total pressure (Pa);P = p/σs0,

dimensionlesspa, Pa asperity pressure (Pa);Pa = pa/σs0p′

c gradient of the film pressure at theentry of the work zone (Pa/mm)

pf , Pf average film pressure (Pa);Pf = pf /σs0

r, r′ undeformed and deformed rollradius (mm)

Rq combined RMS surfaceroughness (mm)

S non-dimensional speed;S = rη0(v0 + vr)/σs0Rqxc

SPi, SPw specific rolling force in the inletzone and work zone (N/mm)

tb, Tb back tension (Pa);Tb = tb/σs0,dimensionless

tf , Tf front tension (Pa);Tf = tf /σs0,dimensionless

vm, vr, vf , v0 average speed, roll speed, flatteningvelocity of asperity, entry speedof the strip (mm/s)

x, X coordinate along the rollingdirection (mm),X = x/xc

xa, Xa entry plane of the inletzone (mm);Xa = xa/xc

xb, Xb point where the asperity contactjust starts (mm);Xb = xa/xc

xc, Xc entry plane of the workzone (mm);Xc = 1

xd exit plane of the work zone (mm)y strip thickness (mm)y0, y1 entry and exit thickness of the

strip (mm)

Greek lettersα pressure coefficient of

viscosity (Pa−1)ε̇ strain rate (s−1)η lubricant viscority (Pa s)

η0 lubricant viscority at atmospherepressure (Pa s)

θa asperity slope (rad)νr νs Poisson rate of the roll and the stripσs, σs0 flow stress and yield strength (Pa)τ, τa, τf total frictional stress and frictional stress

of the asperity and the flow (Pa)φ pressure flow factor

xc

y0

y1

r'

Elastic Inlet Zone Plastic Work Zone

xc

xb

xa

Roll

Strip

Roll

Strip

Asperity

Lubricant

v

xa

tb tf

Lubricant

Elastic inlet zone

v

Zone I Zone II

xy

xd

Fig. 1. Schematic of the rolling process.

outlet zone, the strip is considered to be rigid and we willassume that the contribution of the outlet zone to the asperitydeformation and lubricant film thickness may be neglected.The lubricant pressure falls to zero at the boundary betweenthe work zone and outlet zone.

2.2. Calculation of the flow in the inlet zone

2.2.1. Reynolds equationIn the inlet zone, the lubricant pressure is approximately

governed by the average Reynolds equation[7]:

φh3t

12η

dpf

dx= −vmht + C (1)

where

η = η0 eαpf (2)

vm = 12(vr + v0) (3)

φ is the flow factor which corrects for the influence of rough-ness[7]:

φ = 1 + 3

(Rq

ht

)2

(for zone I) (4)

C. Lu et al. / Journal of Materials Processing Technology 140 (2003) 569–575 571

φ = 2√

3Rq

ht(for zone II) (5)

2.2.2. Lubricant film thicknessAs shown inFig. 1, the lubricant film thickness can be

determined from the shape of the elastic deformed roll plusthe elastic deformationy of the strip:

h(x) = h(xb) +√

r′2 − x2b −

√r′2 − x2 + y(xb) − y(x) (6)

Therefore, at pointxa andxc, the lubrication film thicknessare respectively

h(xa) = h(xb) +√

r′2 − x2b −

√r′2 − x2

a + y(xb) − y(xa)

(7)

h(xc) = h(xb) +√

r′2 − x2b −

√r′2 − x2

c + y(xb) − y(xc)

(8)

In Fig. 1 xc represents the length of the work zone. It doesnot change in this study.

The average film thicknessht then reads[7]

ht = h (for zone I) (9)

ht = (h + √3Rq)2

4√

3Rq(for zone II) (10)

2.2.3. Boundary conditionsBoundary conditions are as follows:

(1) At point xa,

pf (xa) = 0 (11)

dpf (xa)

dx= 0 (12)

(2) At point xb the asperity just contacts the work roll,h =ht. Therefore,Eq. (10)reads

ht(xb) =√

3Rq (13)

(3) At the entry of the work zone (xc), the yield criterionmay be used to obtain

p(xc) = Apa(xc) + (1 − A)pf (xc) = σs0 − tb (14)

In addition, the gradient of the flow pressure will becontinuous at this point because of the continuity of flow.

dpf

dx

∣∣∣∣x=x+

c

= dpf

dx

∣∣∣∣x=x−

c

= p′c (15)

p′c can be obtained from the calculation of the work zone

(seeSection 2.5).

2.2.4. SolutionSubstitution of the boundary condition (Eq. (15)) into

Eq. (1)yields

C = φ(xc)ht(xc)3

12η0 eαpf (xc)p′

c + vmht(xc) (16)

If an initial guess ofxb is given,ht(xc) can be calculated fromEqs. (8) and (10), thenφ(xc) is determined fromEq. (5), p′

cfrom the calculation of the work zone (seeSection 2.5), andpf (xc) from the last iteration. Therefore,C can be obtained.

According to the boundary condition (Eq. (12)) andEq. (1)we know

ht(xa) = C

vm(17)

Moreover,xa can be calculated fromEq. (7).IntegratingEq. (1)and usingEq. (19)yield the film pres-

sure (pf ) and the asperity pressure (pa) in the range of [xc, xa]and [xb, xa], respectively. The total pressure in the entryof the work zone (xc) must satisfy the boundary condition(Eq. (14)). If an error exists,xb will be adjusted by usingthe following equation until convergence is achieved.

xi+1b = xi

b

(σs0 − tb

p(xc)

)n

(18)

wheren is a small positive constant,n = 0.001 is selectedin this study.

2.3. Asperity flattening in the inlet zone

For longitudinal roughness, Wilson and Sheu[10] pro-posed a formulation of the plastic asperity deformation froman upper-bound analysis. The following equation is used tocalculate the asperity pressure in zone II.

pa = 2k

f1E + f2(19)

where

E = ε̇l

vf(20)

f1 = −0.86A2 + 0.345A + 0.515 (21)

f2 = 1

2.571− A − A ln(1 − A)(22)

For wedge shaped asperities of slopeθa, the increase infractional contact area with timet is given by

dA

dt= vf

θal(23)

Substitution ofEq. (20)into Eq. (23)yields

E = 1

θay

dy

dA(24)

In zone II and the plastic work zone, the contact area of thesaw-tooth asperity is decided by the shape of roll and the

572 C. Lu et al. / Journal of Materials Processing Technology 140 (2003) 569–575

elastic deformation of the strip, which is given by

A(x) = 1 −√

ht(x)√3Rq

(25)

2.4. Elastic deformation of the strip in the inlet zone

The elastic deformation of the strip in the inlet zone canbe described by the elastic Von Karman equation presentedby Lugt et al.[4,5], and reads(

Es

νs(1 + νs)

y

y0+ 1 − νs

νsp

)dy

dx+ 1 − νs

νsy

dp

dx− τ = 0

(26)

where

p = Apa + (1 − A)pf (27)

τ = Aτa + (1 − A)τf (28)

τa = mk (29)

τf = ht

2

dpf

dx+ η

v0 − vr

ht(30)

Based on the known total pressure and the frictional stress,Eq. (24)yields the displacement of the strip thicknessy(x)in the inlet zone. Since the hydrodynamic pressure and thepressure gradient are small, the extent of the elastic zonebeyondxa is considered negligible and therefore in this paperxa is also the extent of the elastic zone.

2.5. Work zone analysis

Models regarding mixed lubrication in the work zone havebeen developed by Wilson and Chang[6], Chang and Wilson[7], and Qiu et al.[8]. This study adopted a similar model.The lubricant film pressure can be written as follows

d

dx

(φh3

t

12η

dpf

dx

)= − d

dx(vmht) (31)

Fractional contact area model of asperity reads

dA

dx= − 2x

θar′(2l(1 − A) + yE)(32)

where

E = 2Ak− (p − pa)f2(A)

(p − pa)f1(A)(33)

Relaxation processes was used to improve the convergenceof the Reynolds equation and the asperity calculation.

3. Results and discussion

Parameters used in the calculation are listed inTable 1.Firstly, the program of mixed film calculation in the work

Table 1Parameters used in the calculation

Symbol Value Symbol Value

y0 (mm) 1.0 η0 (Pa s) 0.02y1 (mm) 0.5–0.98 α (Pa−1) 1.58 × 10−8

tb (MPa) 0–68.4 Rq (mm) 0.001tf (MPa) 0–68.4 l (mm) 0.035vr (m/s) 8.57× 10−4–42.8 Er (Pa) 2.1× 1011

σs0 (MPa) 97.75 Es (Pa) 2.1× 1011

r (mm) 200 νr 0.3m 0.2 νs 0.3

0.0 0.2 0.4 0.6 0.8 1.00.0

0.3

0.6

0.9

1.2

1.5

EntryExit

Pf

P

P from Ref.[7] P

f from Ref.[7]

No

n-di

men

sion

al p

ress

ure

P &

Pf

Non-dimensional position X

Fig. 2. Non-dimensional total pressure and film pressure in the work zone.

zone is tested with the examples presented by Chang andWilson [7]. The calculated results are shown inFig. 2. Bothtotal non-dimensional pressureP and non-dimensional filmpressurePf agree well with those from Chang and Wilson[7].

3.1. Elastic deformation of the strip

Two cases, which use the rigid and elastic inlet zonerespectively, are compared. The film thickness and the stripthickness in the inlet zone are plotted inFig. 3. It canbe seen that in the case of the rigid inlet zone the strip

1.000 1.002 1.004 1.006 1.008

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Non-dimensional position X

Non

-dim

ensi

onal

film

thic

knes

s H

t

0.9994

0.9996

0.9998

1.0000

1.0002

Xc

Xr

a Xe

a

Elastic inlet zoneRigid inlet zone

Ht

y

y

Str

ip th

ickn

ess

y, m

m

Fig. 3. Film thickness and the strip thickness in the inlet zone.

C. Lu et al. / Journal of Materials Processing Technology 140 (2003) 569–575 573

1.000 1.002 1.004 1.006 1.0080.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

XcX

ra

Xea

Elastic inlet zone

Rigid inlet zone

Non

-dim

ensi

onal

film

pre

ssur

e P

f

Non-dimensional positon X

Fig. 4. Film pressure in the inlet zone.

thickness remains constant and change sharply at the entryof the work zone. The strip thickness of the inlet zone isreduced in the case of the elastic inlet zone. The throughthickness elastic deformation at pointxc is about 0.43�m.And the change of the strip thickness from the inlet zoneto the work zone is smoother. The average film thicknessdecreases gradually from the entry to the exit of the in-let zone. At the entry of the work zone the average filmthickness for the elastic inlet zone is smaller than the rigidinlet zone by 6.7%. This leads to 0.92% total rolling forcechange in the work zone. The discontinuity of the averagefilm thickness gradient can be seen in the case of the rigidinlet zone because of the discontinuity of the strip thicknessgradient.

Different inlet zone length is observed fromFigs. 3and 4. The length is lengthened in the case of the elasticinlet zone. It can be seen that the influence of the stripelastic deformation on the length of the inlet zone is sig-nificant. In the case of the elastic inlet zone the gradient ofthe asperity change (dA/dx) is smaller than the rigid case.The longer inlet zone is needed so that adequate asperitypressure satisfying the boundary condition (Eq. (14)) can bereached.

In Fig. 4, the film pressure increases with a decrease ofpositionx for both cases. At pointxc the non-dimensionalfilm pressure for the elastic inlet zone is about 0.28, whichis larger than 0.23 in the case of the rigid inlet zone. Thesevalue shows that the asperity carries the majority of load.The gradient of the film pressure is continuous for bothcases because the boundary condition described inEq. (15)is employed.

3.2. Rolling speed

The hydrodynamic pressure of the work zone and in-let zone for various rolling speed are compared inFigs. 5and 6, respectively. The maximum rolling speed calculatedin this study can reach 42 m/s, which exceeds industrialrolling speed in practice. It can be seen that curves crossto each other due to the different inlet zone length and

1.000 1.002 1.004 1.006 1.008 1.010

0.0

0.1

0.2

0.3

0.4

0.5

10-5

10-3

0.01

0.1

0.2

S=0.5

Non

-dim

ensi

onal

film

pre

ssur

e P

f

Non-dimensional position X

Fig. 5. Hydrodynamic pressure in the work zone.

the hydrodynamic pressure gradient for the different rollingspeed. The length of the inlet zonexa decreases and the gra-dient of the hydrodynamic pressurePf increases with therolling speed. The high hydrodynamic pressure at the entryof the work zone is produced at high rolling speeds. In thework zone, the maximum pressure increases with an increas-ing rolling speed at a lower value ofS (S < 10−2). Thenthe maximum pressure will be decreased and the flatteningpressure top can be observed at a higher value ofS (S ≥10−2). This is because the hydrodynamic pressure increaseswith the rolling speed until it reaches the total pressure.The increased hydrodynamic pressure decreases the contactarea and reduces the friction, which results in a lower totalpressure.

Fig. 7 shows the film thickness and the length of the in-let zone for different roll speed. Results of two models arecompared. All models show that the film thickness increaseswith an increase of the rolling speed. Increasing the rollspeed leads to an increase of the film pressure in the inletzone. To meet the yield condition at the entry of the workzone, the pressure that the asperity carries is reduced. There-fore, a short inlet zone length and large film thickness areproduced. Wilson and Walowit’s results change sharply withan increase of the speed and a large value ofHt(xc) is pro-duced at high speed (S > 0.1). This means that the rolling

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.2

0.50.10.01

10-3

10-4

S=10-5

EntryExitNon

-dim

ensi

onal

film

pre

ssur

e P

f

Non-dimensional position X

Fig. 6. Hydrodynamic pressure in the inlet zone.

574 C. Lu et al. / Journal of Materials Processing Technology 140 (2003) 569–575

1E-4 1E-3 0.01 0.1 0.5

0.0

0.4

0.8

1.2

1.6

Ht(x

c)

xa

Ht(x

c)

Non-dimensional roll speed S

Non

-dim

ens

ion

al f

ilm th

ickn

ess

Ht(x

c)

This study, with strip elastic deformation Wilson and Walowit [3]

0.040

0.045

0.050

0.055

0.060

0.065

0.070

Inle

t zon

e le

ngth

xa,

mm

Fig. 7. Film thickness and the inlet zone length at different roll speed.

speed in the full film regime used by Wilson and Walowit’smodel has a more significant effect on the film thicknessthan the mixed film regime in this study. AsS = 0.2, thenon-dimensional film thickness of this study and Wilson andWalowit’s model at pointxc are 0.85 and 1.23, respectively.The difference will produce 3.94% total rolling force changein the work zone.

Wilson and Walowit[3] assumed that the hydrodynamicpressure is zero at the infinite distance beyond the work zone.This means that the assumed length of the inlet zone is avery large value. However, the inlet zone length can be deter-mined by using the appropriate boundaries (Eqs. (11)–(15))in this study. The dot line represents the length of the in-let zone in theFig. 7. It decreases as the rolling speed isincreased.

3.3. Reduction

The film thickness and the inlet zone length at differentreductions are shown inFig. 8. The film thickness at pointxc decreases with an increase of reduction. The change is

0 10 20 30 40 500.0

0.5

1.0

1.5

2.0

2.5

3.0This study, with strip elastic deformation Wilson and Walowit [3]

xa

Ht(x

c)

Reduction, %

Non

-dim

ens

iona

l film

thi

ckne

ss H

t(xc)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Inle

t zon

e le

ngth

xa,

mm

Fig. 8. Effect of the strip elastic deformation at entry on the film thicknessat different reduction.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Ht(x

c)

Ht(x

c)

Non-dimensional back tension Tb

Non

-dim

ensi

onal

film

thic

knes

s H

t(xc)

This study, with strip elastic deformation Wilson and Walowit [3]

0.01

0.02

0.03

0.04

0.05

0.06

xa

Inle

t zon

e le

ngth

xa,

mm

Fig. 9. Film thickness at different back tension.

relatively large for the reduction less than 10%. But thefilm thickness calculated by the full film regime (Wilsonand Walowit’s model) changes sharply as the reduction isincreased. Similar trend can be observed for the inlet zonelength because the high reduction produces a sharp changeof asperity and a short inlet zone.

3.4. Tension

From Fig. 9 it can be seen that the back tension has asignificant influence on the film thickness and the inlet zonelength. As the back tension increases, the film thickness(Ht(xc)) at the entry of the work zone increases. The changeof the film thickness in the mixed film regime (this study)is larger than the full film regime (Wilson and Walowit’smodel). This indicates that the mixed film regime is moresensitive to the back tension. The inlet zone length has areverse trend. The application of the back tension reduces thetotal pressure at the yield point (xc) (Eq. (14)). A short inletzone is needed to satisfy the yield criterion. The asperity hasa small deformation in the inlet zone and a small contactarea at point (xc). Therefore, the film thickness at point (xc)increases with an increase of the back tension.

4. Conclusions

A mixed film lubrication model in the inlet zone has beendeveloped. Elastic deformation of the strip has been incor-porated in the model. The influence of the elastic deforma-tion of the strip, reduction, roll speed and tension on thelubricant film thickness and the inlet zone length has beenstudied. From the results of the numerical simulations thefollowing conclusions can be obtained:

• Elastic deformation of the strip enlarges the length of theinlet zone. The lubricant film thickness (Ht(xc)) at theentry of the work zone is higher than that of the rigid case.

• The rolling speed, reduction and back tension have a sig-nificant influence on the lubrication film thicknessHt(xc)and the inlet zone length.

C. Lu et al. / Journal of Materials Processing Technology 140 (2003) 569–575 575

• The mixed film model shows a large difference with theresults of the full film model.

Acknowledgements

The authors gratefully acknowledge the financial supportof the research from an Australia Research Council Discov-ery Grant.

References

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