SOME ISSUES ARISING IN FINDING THE

DETACHMENT POINT IN CALENDERING OF

PLASTIC SHEETS**

Evan Mitsoulis*School of Mining Engineering & Metallurgy, National Technical

University of Athens, Zografou, 157 80 Athens, Greece

ABSTRACT: Calendering is a process for producing plastic sheets of a desiredfinal thickness and appearance. The thickness of the exiting sheet during acalendering operation is uniquely found by the lubrication approximation theoryand the application of the Swift boundary conditions, which dictate that both thepressure and its axial derivative are zero at detachment. This cannot be used in a2D analysis of the process, where the detachment point is the anchor of a freesurface, and hence a singular point where both the pressure and the stresses gothrough numerical oscillations. This difficulty can be circumvented by using theboundary element method (BEM), which uses as primary variables velocities andtractions, and thus avoids pressures and stresses. Then the detachment point isfound as the point where the tangential traction becomes zero. Numerical testsundertaken here with the finite element method (FEM) show that the LATresults can be used as a good approximation for the detachment point, which isthen fixed. Comparisons with 2D BEM results show a good agreement for allflow field variables. However, the exact position of the detachment point in a 2DFEM analysis is still elusive, since for viscous polymer melts the contact angle isnot known and should be part of the solution. Some thoughts are given abouthow to tackle this still unresolved issue, based on double nodes withdiscontinuous velocities and pressures.

KEY WORDS: calendering, power-law model, sheet thickness, detachmentpoint, finite elements, free surfaces.

*E-mail: mitsouli@metal.ntua.gr**Dedicated to the memory of Jim Harrington.Figures 1–4, 7–11 and 15 appear in color online: http://jpf.sagepub.com

JOURNAL OF PLASTIC FILM & SHEETING, VOL. 26—APRIL 2010 141

8756-0879/10/02 0141–25 $10.00/0 DOI: 10.1177/8756087910376144� The Author(s), 2010. Reprints and permissions:http://www.sagepub.co.uk/journalsPermissions.nav

INTRODUCTION

CALENDERING IS A process used in many industries, such as the paper,plastics, and rubber industries, for the production of rolled sheets of

specific thickness and final appearance. The process is shown schema-tically in Figure 1, where a sheet of initial thickness AS is taken up by twoco-rotating rolls (calenders) to form a sheet of final thickness EF [1]. Dueto the forced entry of a thick sheet through a small gap, a part of thematerial flows back. Thus, there are recirculation patterns forming in theentry region. Another feature of the process is the formation of freesurfaces before and after passing through the nip region (at entry ABC,and at exit DE and GF). Their determination presents extra difficulties,due to their unknown a priori location and shape.

The process has been extensively studied by many researchers overthe last 50 years. Starting with Ardichvili [2] in 1938, the LubricationApproximation Theory (LAT) of Reynolds was employed, and usefulresults were obtained due to the ingenious Swift condition, which statesthat the pressure and its gradient are zero at detachment of the sheetfrom the rolls [1]. All works based on LAT have been reviewed by Sofouand Mitsoulis [3], who also employed LAT for viscoplastic (Bingham orHerschel–Bulkley) materials. The LAT analysis was followed by 2Dstudies for PVC sheets [4,5], viscoelastic fluids [6], and viscoplasticmaterials [7], using numerical methods such as the finite elementmethod (FEM) [4,5,7] and the boundary element method (BEM) [6]. Asingle 3D study based on FEM has also appeared in the literature [8].

What happens exactly at the attachment and detachment points(points C and D in Figure 1) is not clear-cut, since these are singularpoints where the pressure and the stresses tend to infinity, and in anyFEM solution there appear high oscillations (zig-zags). The problem wasfirst addressed in calendering by Zheng and Tanner [6], who commentedon that and then proceeded to use the BEM having the velocities andtractions as primary variables. They concentrated only on the detach-ment point and they used the criterion that the fluid detaches when the

S

D

G F

EExit

AB

C

Entry

Figure 1. Free surfaces in calendering. The incoming plastic sheet enters with thicknessAS and exits with thickness EF, forming the free surfaces ABC, DE, and GF.

142 E. MITSOULIS

tangential traction goes through zero. This way they were able tocircumvent the problem of pressure and stresses, which oscillate withhigh peaks in a FEM velocity–pressure formulation.

The recent 2D analysis by the author [7] uses the results from LAT asa good enough approximation to the location of the attachment anddetachment points. The purpose of the present paper is to examine howgood such an approximation is, and the way to do this is to examine theresults in comparison with those by Zheng and Tanner [6]. Finally, somedetails are given for a more thorough but also more difficult tackling ofthe problem within the FEM context.

MATHEMATICAL MODELING

Governing Equations

The flow in calendering is well described in the textbook byMiddleman [1]. It is governed by the well-known conservation equationsof mass, momentum, and energy, and a constitutive equation thatdescribes the relation between the stresses,���, and the velocity gradients,r �v. Usually, the generalized Newtonian fluid is assumed with a viscositythat depends on the magnitude of the rate-of-strain tensor. In fulltensorial form the constitutive equation (Equation (1)) is written as:

��� ¼ � ��_�, ð1Þ

Where:

� ¼ apparent viscosity (Equation (2)) given by the power-law model [1]:

� ¼ Kj _�jn�1: ð2Þ

In the above,

K ¼ consistency index

n ¼ power-law index

| _�| ¼ rate-of-strain tensor magnitude, ��_g ¼ r�vþr�vT , which is given by Equation (3):

j _�j ¼

ffiffiffiffiffiffiffiffiffiffi1

2II _�

r¼

1

2f��_� :

��_�g

� �1=2

, ð3Þ

Where:

II _� ¼ second invariant of ��_�.

Detachment Point in Calendering of Plastic Sheets 143

In calendering the following dimensionless parameters (Equation (4))are introduced [1]:

x0 ¼xffiffiffiffiffiffiffiffiffiffiffiffiffi

2RH0

p , y0 ¼y

H0, h0 ¼

h

H0¼ 1þ

x2

2RH0, P0 ¼

P

K

H0

U

� �n

,

�2 ¼Q

2UH0� 1, ð4Þ

Where:

� ¼ dimensionless flow rate (or leave-off distance)

Q ¼ flow rate

R ¼ roll radius

h ¼ local height

and the rest of the symbols are defined in Figure 2, which presents thecase for symmetric calendering as in Zheng and Tanner [6] andMitsoulis [7].

For a desired value of �, the exiting sheet thickness H is given by(Equation (5)):

H

H0¼ 1þ �2, ð5Þ

while the thickness (Equation (6)) of the incoming sheet Hf is enteringthe analysis according to the definition:

x0f ¼Hf

H0� 1

� �1=2

: ð6Þ

2Hf

yx0

H0

A

U

B

CD

EF

−xf

l

–l

2H

x

Figure 2. Flow field and notation for the 2D analysis of calendering (symmetric case). Thesheet enters from left with thickness 2Hf at point E (given), detaches at point D (or leave-offdistance �) (to be found), and exits with thickness 2H at point C (also to be found).

144 E. MITSOULIS

Operating Variables

The operating variables used in engineering calculations are also ofinterest [1]:(i) the maximum pressure (Equation (7)), PðnÞ, defined by:

PðnÞ ¼P0maxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2R=H0

p ¼2nþ 1

n

� �nZ �

��

IðnÞdx0, ð7Þ

(ii) the roll-separating force (Equations (8a) and (8b)) per unit width,F=WðnÞ, defined by:

F

WðnÞ ¼

Z �

�x0f

PðxÞdx ¼ KU

H0

� �n

RFðnÞ, ð8aÞ

with FðnÞ ¼ 22nþ 1

n

� �nZ �

�x0f

Z �

x0IðnÞdx0

� �dx0, ð8bÞ

(iii) the torque (Equation (9)) for each roll, TðnÞ, defined by:

TðnÞ ¼ WR

Z �

�x0f

�xy

��y¼hðxÞ

dx, ð9Þ

(iv) the power input (Equations (10a) and (10b)) for both rolls, _WðnÞ,defined by:

_WðnÞ ¼ 2WU

Z �

�x0f

�xy

��y¼hðxÞ

dx ¼ WU2KU

H0

� �n�1ffiffiffiffiffiffiffiR

H0

sEðnÞ, ð10aÞ

with EðnÞ ¼ �2ffiffiffi2p 2nþ 1

n

� �nZ �

�x0f

IðnÞ 1þ x02�

dx0: ð10bÞ

Note that (Equation (11)):

_WðnÞ ¼ 2U=RTðnÞ: ð11Þ

In the above the integrand I(n) is given in Sofou and Mitsoulis [3].

Detachment Point in Calendering of Plastic Sheets 145

Boundary Conditions

The constitutive equations for power-law fluids are solved togetherwith the conservation equations and appropriate boundary conditions.Figure 2 shows the flow domain and the boundary conditions for thesymmetric problem. Due to symmetry, only one half of the flow domainis used for the solution, as was done previously [6,7].

Referring to Figure 2, the plastic sheet comes into first contact withthe rolls at the entry point E (dimensionless entry distance, �x0f ), withinitial thickness 2Hf (at point E, but not necessarily at point F). It leavesthe calendering system at the detachment point D (or dimensionlessleave-off distance, �), while it takes the final thickness 2H at point C. Incalendering, one of the two points of attachment or detachment isunknown, and finding the other is part of the solution.

Then, the boundary conditions are the following (flow from left toright):

. symmetry along AB (vy¼ 0, �xy¼ 0);

. no-slip at the roll surface ED (tangential velocity vt¼U, normalvelocity vn¼ 0);

. along the free surface of the exiting sheet, DC, the tangential andnormal stresses are zero ðð ��� � �nÞ � �t ¼ 0,ð ��� � �nÞ � �n ¼ 0), and there is noflow through it (�v � �n ¼ 0),

. where: �n and �t are the normal to the surface and tangential unitvectors, respectively, and ��� ¼ �p

��I þ ��� are the total stresses;. along the exit boundary BC, the tangential and normal stresses are

zero ðð ��� � �nÞ � �t ¼ 0,ð ��� � �nÞ � �n ¼ 0);. along the entry boundary AF, the tangential and normal stresses are

zero ðð ��� � �nÞ � �t ¼ 0,ð ��� � �nÞ � �n ¼ 0);. along the free surface of the entering sheet, FE, the tangential and

normal stresses are zero ðð ��� � �nÞ � �t ¼ 0,ð ��� � �nÞ � �n ¼ 0), and there is noflow through it (�v � �n ¼ 0).

The reference pressure is set to 0 at point B.All lengths are scaled by the minimum half gap, H0, all velocities by

the roll speed, U, and all pressures and stresses by K(U/H0)n.

METHOD OF SOLUTION

The numerical solution is obtained with the FEM, using the programUVPTH, originally developed for multilayer flows [9,10], which employsas primary variables the two velocities, pressure, temperature, and free

146 E. MITSOULIS

surface location (u-v-p-T-h formulation). It uses 9-node Lagrangianquadrilateral elements with biquadratic interpolation for the velocities,temperatures and free surface location, and bilinear interpolation forthe pressures. The free surface is found in a coupled way as part of thesolution for the primary variables.

In the present work we have selected two geometries: (a) calendering-1with aspect ratio R/H0¼ 100, and (b) calendering-2 with aspect ratio R/H0¼ 1000. The first case refers to the work by Zheng and Tanner [6], withthe purpose of setting the detachment point from the rolls and comparingthe corresponding results for Newtonian and power-law fluids. Thesecond case refers to previous Newtonian simulations by Mitsoulis et al.[4], and corresponds to ‘strong’ calendering due to the large aspect ratio.Data for the two geometries are given in Table 1.

For each of the geometry we have used two meshes, which are shownin Figure 3 for the case of calendering-1 (Figure 3(a)) and calendering-2(Figure 3(b)). The meshes of Figure 3(a) are shown after the Newtoniansolution has been achieved; hence the deformed domain due to the freesurfaces at entry. Those of Figure 3(b) are shown undeformed as theyare at the beginning of the solution process.

In each geometry mesh M1 (lower half) has 2058 elements, while meshM2 (upper half) has 8232 elements, and is produced by subdividing eachelement of M1 into 4 sub-elements. Mesh M2 gives 33,573 nodes and74,920 unknown degrees of freedom (DOF). The less dense mesh M1with 2058 elements was used primarily for preliminary runs to gainexperience with 2D calendering flows.

As explained in detail in Mitsoulis [7], many tests and runs have beenmade for finding points E and D, setting one and looking for the other.However, there was no criterion for uniqueness of solution, since both ofthese points are singular points, and both the pressure and the shearstress showed discontinuities there in the form of oscillations (zig-zags).To get a solution which was not far from reality, it was found best to set theentry point, �x0f , and detachment point, �, using the lubricationapproximation (LAT) [3].

Table 1. Geometry data used in the simulations of calendering.

GeometryRadius,R (cm)

Minimum gap,2H0 (cm)

Aspect ratio,R/H0 (–)

Entry thickness,Hf/H0 (–)

Calendering-1 25 0.5 100 20Calendering-2 10 0.02 1000 23.5625

Detachment Point in Calendering of Plastic Sheets 147

More precisely, for a given attachment point, �x0f (hence a given sheetthickness 2Hf) the detachment point, �, is found using a software basedon LAT [3]. Then, the extent of the flow domain (entry point F, point ofattachment E and point of detachment D, and exit point C from the flowfield) are the input to the 2D software used in the 2D simulations. PointsF and C are then found as part of the free surfaces FE and DC, with thepoints E and D being set as anchors from where the integrations for thefree surfaces are started. The unknown thicknesses AF and BC are thusfound and so are the unknown a priori entry and exit sheet velocities.This method allows for a 2D analysis of the problem without significanterrors, as will be shown below, at least for calendering cases with a largeaspect ratio R/H044 1.

As shown in Figure 3, the meshes are denser in the regions near theattachment and detachment points, �x0f and �, and the antisymmetricpoint, –�. This happens because at these points the extrema for pressureoccur, as well as the steepest changes according to LAT. More precisely,

x/H0

x/H0

y/H

0y/

H0

–280 –240 –200 –160 –120 –80 –40 0 40

–120–50

–40

–30

–20

–10

0

10

20

–20

0

20

30

40

50

–100 –80 –60 –40 –20 0

(a)

(b)

Figure 3. Finite element meshes used in the computations. Upper half shows a meshwith 8232 elements (M2), lower half mesh has 2058 elements (M1): (a) calendering-1(deformed mesh from Newtonian solution) and (b) calendering-2 (undeformed mesh at thebeginning of computations).

148 E. MITSOULIS

the pressure is zero at attachment and detachment, while it takes itsmaximum value at –�.

The mesh density at detachment is shown in Figure 4, where manyelements are concentrated before and after the detachment point D, asfound from LAT. From this point on (also at the correspondingattachment point E), the free surface starts, and there is a change inthe boundary conditions from no-slip at the wall (given velocity) to a freesurface with unknown velocity. The assumption made in LAT that thevelocity at the detachment point and thereafter is equal to the roll speedis not valid for the fully 2D analysis, as will be shown below.

Considering the entry and exit lengths for the incoming and outgoingsheets, the flow domain entry is set at –3Hf and the exit at 4H, so thatenough length is given for a fully developed velocity profile at entry andexit. Knowing that calendering is mainly a process dominated by shearand that plastic sheets such as PVC sheets are mainly pseudoplasticfluids, which in contrast to viscoelastic fluids, do not swell very much[11,12], these lengths are considered adequate for a full rearrangement ofthe profiles. This adequacy was checked for each case by plotting the axialprofiles for u and v (Figure 5 for calendering-1 of a Newtonian fluid)along the roll surface (wall, w), at the symmetry line (centerline, cl) andon the free surface (fs), and observing their leveling off (plug profile).

This leveling-off criterion is better shown in Figure 6, where theu-profiles are shown at the symmetry line (cl) and on the free surface(fs): (a) at entry and (b) at exit of the flow field. We observe that the

6.5 7 7.5 865.55

1.4

1.2

1

0.8

0.6

0.4

0.2

0

x/H0

y/H

0

Figure 4. Blown-up FEM mesh near the separation point at exit (deformed mesh fromNewtonian solution) in calendering-1. Note the small swelling of the free surface afterdetachment.

Detachment Point in Calendering of Plastic Sheets 149

velocities at symmetry line and free surface coincide, both at entry andat exit of the flow field. This guarantees that the flow fieldextent upstream and downstream from the nip region is adequate, sothat a plug velocity profile is obtained for the incoming and outgoingsheet.

x „=x/H0 x „=x/H0

(a) (b)

–120 –110 –100 –90 –80 –70 –60 –50

Vel

ocity

, u′=

u / U

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

fs

cl

5 6 7 8 9 10

Vel

ocity

, u′=

u / U

0.95

1.00

1.05

1.10

1.15

fs

cl

w

Newtonian fluid l=0.4712

Newtonian fluid l=0.4712

Leave-off distance, l

Entry point, −xf

Figure 6. Calendering-1 of a Newtonian fluid (Table 1). Blow-up of the axial velocityprofile, u: (a) in the entry and (b) in the exit region of the flow field. The profiles level off atthe ends showing the adequacy of the flow field for the development of the free surfacesupstream and downstream of the nip region.

x„=x /H0 x„=x /H0

–120 –100 –80 –60 –40 –20 0

Vel

ocity

, u„=

u / U

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4Newtonian fluid l=0.4712

Entry point, −xf

fs

w

Leave-off distance, l

cl

–120 –100 –80 –60 –40 –20 0V

eloc

ity, v

„=v

/ U

–0.6

–0.4

–0.2

0.0

0.2

fs

w

cl

(a) (b)

Entry point, −xf

Newtonian fluid l=0.4712

Leave-off distance, l

Figure 5. Calendering-1 of a Newtonian fluid (Table 1). Development of velocity profilesin the direction of flow: (a) axial velocity, u, and (b) tranverse velocity, v. The profilescoincide at entry and exit of the flow field, guaranteeing the adequacy of the entry and exitlengths for the development of the free surfaces upstream and downstream of the nipregion.

150 E. MITSOULIS

RESULTS AND DISCUSSION

Newtonian Fluids

We start with the Newtonian results for the two calendering cases.Our previous work [7] gave evidence that a comparison of resultsbetween LAT and FEM showed that using LAT for finding thedetachment point gives very good predictions in comparison withthose found by the 2D analysis of Zheng and Tanner [6] with boundaryelements. Any other effort was not able to give such good results.

Here we present results for several flow variables obtained from theFEM analysis. In calendering-1, the aim is to compare our results withthose of Zheng and Tanner [6] to find out any differences in the twomethods of solution (FEM vs. BEM). In calendering-2, the aim is to showthe effect of geometry on the flow variables.

Figure 7 shows contours of the flow variables for calendering-1. For everyvariable, 11 contours are drawn between the maximum and minimumvalues. Figure 7(a) shows the streamlines, that is, contours of the streamfunction, . The latter has been normalized (Equation (12)) according to:

� ¼ � cl

C � cl, ð12Þ

where:

* ¼ dimensionless normalized value

¼ actual value

cl ¼ centerline value at exit B (¼0)

C¼HCUC ¼ value at the free surface at exit.

Thus, *¼ 1 corresponds to half the flow rate that exits ( C¼Q/2)due to symmetry. Any negative value of * corresponds to recirculation(vortex development), with *v,max being the maximum value of thevortex in the flow field.

It is observed that at the entry of the flow field there are tworecirculation regions. The two vortices are symmetric, as expected, bothin size and strength, and their center is located at the attachmentdistance, �xf, with intensity *v,max¼�1.91, which means that 191% ofthe fluid recirculates.

Figure 7(b)–(e) shows the pressure (isobars) and the stresses (isostresscontours), which have been made dimensionless by the term �U/H0. Weobserve that the pressure (Figure 7(b)) takes its maximum value at

Detachment Point in Calendering of Plastic Sheets 151

30

20

10

0

–10

–20

–3030

20

10

0

–10

–20

–3030

20

10

0

–10

–20

–3030

20

10

0

–10

–20

–3030

20

10

0

–10

–20

–30–120 –100 –80 –60 –40 –20 0

–0.66–0.320.000.010.020.030.080.130.200.290.60TXY

–0.93–0.15–0.07–0.05–0.03–0.01

0.000.010.020.050.65TYY

–0.24–0.15–0.05–0.03–0.02–0.010.010.030.050.120.75TXX

–0.76–0.14

0.471.091.702.322.943.554.174.795.40

P

–1.91–1.62–1.33–1.04–0.75–0.46–0.16

0.130.420.711.00STR

x/H0

y/H

0y/

H0

y/H

0y/

H0

y/H

0

(a)

(b)

(c)

(d)

(e)

Figure 7. Contours in calendering-1 of Newtonian fluids. Flow variables: (a) streamfunction, (STR), (b) pressure, P (P), (c) �xx (TXX), (d) �yy (TYY), and (e) �xy (TXY). Thestream function has been normalized by the flow rate Q, while the pressure and stresseshave been made dimensionless by the term �U/H0.

152 E. MITSOULIS

distance ��, as predicted by LAT, and decreases rapidly towards theexit, where it becomes zero. Regarding the normal stresses, we observethat �xx (Figure 7(c)) increases from the fluid entry up to the nip region,and from the centerline up to the roll surface, while the opposite occursfor �yy (Figure 7(d)). It is noted that from the continuity equation for 2Dincompressible flows, we have �yy¼��xx. Any differences observedbetween Figure 7(c) and (d) are due to different choices in the contourvalues. At entry point, �xf, and at the roll surface where the fluid firstcomes into contact with the rolls, there are interesting contours of shearstress (Figure 7(e)). This is due to the singular nature of the attachmentpoint, as calculated by FEM.

Zheng and Tanner [6] have given contour results for the streamfunction (STR) and the velocity components u (U) and v (V). Figure 8presents results corresponding to those in Zheng and Tanner [6]. Thedimensionless (non-normalized) value of the stream function (Figure8(a)) found by FEM here is 0.3 (¼HCUC), and is identical to the valuegiven by the BEM results. The bigger differences are observed in theentry of the incoming sheet, where Zheng and Tanner [6] ignored thefree surface, thus cutting the flow field there. This in turn had the effectof not allowing the full development of the recirculating vortices, whichled to a nonsmooth closing of the streamlines and the other contours atentry. However, if one is interested in what is happening in the nipregion, the results do not differ appreciably.

Figure 8(b) and (c) present the results for the velocities u, v,respectively, made dimensionless by the roll speed. The maximumvalue of u is 1.33, while Zheng and Tanner [6] give 1.3, and this occurs,as expected, in the nip region. The extrema for v are found at the twosingular points where the material first comes into contact with therolls. The velocity contours show the same behavior as those found byBEM [6].

The presentation of the results is continued in Figure 9 with the caseof ‘strong’ calendering-2 with a ratio R/H0¼ 1000. The contours havethe same behavior as in calendering-1. Some differences worth notingare the larger flow field, which leads to bigger vortices with intensity *v,max¼�2.56, which means that 256% of the fluid recirculates.Similarly, the other variables have higher absolute values, as shown inTable 2, where the results from both cases of calendering are puttogether (C-1 and C-2). It is interesting to note that the swelling incalendering-2 is still smaller (0.5% vs. 1.6% for C-1) and that thevelocities at the exit are close to the roll speed, but somewhat smaller. Itshould be noted that LAT assumes that the exit speed is equal to the rollspeed U.

Detachment Point in Calendering of Plastic Sheets 153

Power-law Fluids

We present results for pseudoplastic fluids obeying the power law forthe case of calendering-1. The purpose here is to study the effect of thepower-law index on the various quantities. Tables 3 and 4 list thenumerical results from LAT and FEM for comparison. The subscripts F,C, B refer to the boundary points of Figure 2. A reduction in the power-law index leads to an increase of the entry thickness with a simultaneousdecrease of the entry speed. An increase of pseudoplasticity (loweringof n) leads to higher values for the leave-off distance, �, and to a smallerswelling.

Listed in Table 4 are the values of the dimensionless operatingvariables (Equations (7)–(9)) for pseudoplastic fluids obtained both from

x/H0

–30

–20

–10

0

10

20

30

y/H

0

STR1.000.710.420.13

–0.16–0.46–0.75–1.04–1.33–1.62–1.91

x/H0

x/H0

–30

–20

–10

0

10

20

30y/

H0

U1.331.160.990.820.640.470.300.13

–0.04–0.21–0.38

–120 –100 –80 –60 –40 –20 0

–120 –100 –80 –60 –40 –20 0

–120 –100 –80 –60 –40 –20 0–30

–20

–10

0

10

20

30

y/H

0

V0.600.480.360.120.020.00

–0.02–0.12–0.36–0.48–0.60

Calendering–1, this workZ&T(a)

(b)

(c)

Figure 8. Comparison between results for calendering-1 of Newtonian fluids by Zhengand Tanner [6] (Z&T) and of this work. Contours of flow variables: (a) stream function, (STR), (b) axial velocity u (U), and (c) transverse velocity v (V). The stream function hasbeen normalized by the flow rate Q, while the velocities have been made dimensionless bythe roll speed.

154 E. MITSOULIS

40

20

0

–20

–4040

20

0

–20

–4040

20

0

–20

–4040

20

0

–20

–4040

20

0

–20

–40

STR1.000.640.29

–0.07–0.43–0.78–1.14–1.49–1.85–2.21–2.56

P17.0415.3013.5711.8310.096.623.141.410.370.01

–0.33

TXX0.670.080.060.050.030.010.00

–0.01–0.02–0.03–0.06

TYY0.140.050.030.020.01

–0.00–0.01–0.02–0.03–0.05–0.74

TXY0.620.320.220.160.120.090.040.030.030.01

–0.68

400–40–80–120–160–200–240–280

y/H

0

x/H0

y/H

0y/

H0

y/H

0y/

H0

(a)

(b)

(c)

(d)

(e)

Figure 9. Contours in calendering-2 of Newtonian fluids. Flow variables: (a) streamfunction, (STR), (b) pressure, P (P), (c) �xx (TXX), (d) �yy (TYY), and (e) �xy (TXY). Thestream function has been normalized by the flow rate Q, while the pressure and stresseshave been made dimensionless by the term �U/H0.

Table 2. The 2D results from calendering of Newtonian fluids.

Geometryk

(LAT)Entry thickness

HF/H0

Entry speedUF/U

Exit thicknessHC/H0

Exit speedUC/U

Exit swellHC/HB (%)

C-1 0.4712 11.82 0.073 1.242 0.981 1.60C-2 0.4720 9.61 0.083 1.229 0.995 0.51

Detachment Point in Calendering of Plastic Sheets 155

LAT and FEM. There is a very good agreement between the twomethods regarding the pressures and the roll-separating force.Discrepancies are found in the values of the power coefficient, E (seeFigure 14 and discussion further below).

Results are presented for the flow field of pseudoplastic fluids incalendering-1. The values of the power-law index are given in Table 4.Figure 10 presents the streamlines. We observe that at entry to the flowfield two recirculation regions are formed. A reduction in the power-lawindex affects both the location of the eye of the vortex and its intensity.The eye of the vortex for the Newtonian fluid (Figure 10(a)) is located atthe entry distance, �xf, with intensity *v,max¼�1.91, which meansthat 191% of the fluid recirculates. A reduction in the power-law indexleads to a gradual movement of the eye of the vortex towards the nip. Italso brings about a gradual reduction in the vortex intensity (see Table 4for *v,max), and leads to a final disappearance of the vortex at aboutn¼ 0.2. This phenomenon is due to the very high shear-thinning natureof the fluid when the power-law index is quite low.

Table 4. Results from calendering-1 for pseudoplastic fluids obeying thepower law, using LAT and FEM.

Maximum pressure,P (–)

Roll-separatingforce, F (–)

Power,E (–)

n k (LAT) *�, max LAT FEM LAT FEM LAT FEM

1.0 0.471 �1.910 0.372 0.382 1.082 1.085 2.765 2.6000.8 0.473 �1.614 0.464 0.473 1.371 1.371 3.607 3.3250.6 0.476 �1.198 0.578 0.583 1.736 1.734 4.707 4.2580.4 0.478 �0.653 0.717 0.712 2.185 2.178 6.106 5.4200.2 0.480 �0.008 0.869 0.849 2.681 2.656 7.718 6.636

Table 3. Results from calendering-1 for pseudoplastic fluids obeying thepower law, using LAT (for finding k) and FEM.

nk

(LAT)Entry thickness

HF/H0

Entry speedUF/U

Exit thicknessHC/H0

Exit speedUC/U

Exit swellHC/HB (%)

1.0 0.471 11.82 0.073 1.242 0.981 1.600.8 0.473 13.96 0.073 1.241 0.981 1.410.6 0.476 17.99 0.062 1.241 0.980 1.220.4 0.478 24.87 0.035 1.237 0.978 0.910.2 0.480 26.73 0.025 1.249 0.973 0.46

156 E. MITSOULIS

30

20

10

0

–10

–20

–3030

20

10

0

–10

–20

–3030

20

10

0

–10

–20

–3030

20

10

0

–10

–20

–3030

20

10

0

–10

–20

–30–120 –100 –80 –60 –40 –20

1.00

1.000.830.670.500.340.170.01

–0.16–0.32–0.49–0.65

1.00

1.00STR

0.740.480.22

STR

0.780.560.340.12

–0.10

–1.61

–1.91–1.62–1.33–1.04–0.75–0.46–0.160.13

0.420.71

1.00STR

–1.35–1.09–0.83–0.57–0.31–0.05

–0.32–0.54–0.76–0.98–1.20

STR

STR

0.900.800.700.600.500.400.310.210.110.01

0x/H0

y/H

0y/

H0

y/H

0y/

H0

y/H

0

(a)

(b)

(c)

(d)

(e)

Figure 10. Results for the streamlines in calendering-1 of power-law fluids for differentvalues of the power-law index: (a) n¼ 1, (b) n¼ 0.8, (c) n¼ 0.6, (d) n¼ 0.4, and (e) n¼ 0.2.The stream function, , has been normalized by the flow rate Q.

Detachment Point in Calendering of Plastic Sheets 157

More specifically, moving away from the Newtonian behavior wherethe viscosity is constant all over the flow field, the viscosity increases inregions of small strain rates such as those in the entry, and thus theability of the fluid to rotate is reduced. This becomes more obvious bylooking at the shape of the free surface at entry. As shown in Figure10(a), the height of the free surface of the Newtonian fluid upstream ismarkedly different from the height at the incoming plane, and the freesurface has a high curvature.

As the power-law index decreases, the fluid comes into contact with alarger part of the roll surface, and the curvature of the free surface isreduced due to the increase of the viscosity in this region. At the extremecase of pseudoplasticity (n¼ 0.2, Figure 10(e)), the thickness upstream isabout equal to the entry thickness, and the percentage of the fluid whichrotates is almost zero.

Figure 11 presents a comparison between the results of Zheng andTanner [6] (Z&T) and those from the present work for calendering-1 ofpseudoplastic fluid with n¼ 0.4. Figure 11(a) presents the streamlines(STR) and Figure 11(b), and (c) the velocity components u (U) and v (V).

It is obvious that the results by Zheng and Tanner [6] have the samedifficulties for the streamlines as the corresponding Newtonian ones ofFigure 8, because of the lack of a free surface in the entry region.Namely, we observe nonphysical distortions of the streamlines and ofthe velocity contours, and this led the authors to conclude that the non-Newtonian vortex is larger than the Newtonian one, which is contrary tophysical arguments. Obviously, not solving for the free surface leads toerrors for the flow behavior, but not for the dynamic variables which aredetermined primarily by what is happening between the rolls.

Figure 12 presents the results for the pressure distribution, whichhave also been given in [6].

The pressure values have been made dimensionless by dividing by themaximum Newtonian value and that is why they decrease as ndecreases, in contrast with the results of Table 4, where the increaseis due to a dimensionalization that takes into account the value of n.There is agreement between the two works regarding the ordering of thecurves, while discrepancies are found in the shape of the curves at theentry to the flow field. This is due to the different methods of solution.Zheng and Tanner [6] did not take into account the free surfaceupstream, so their flow field is shorter. This causes the pressure tendingto zero in a shorter distance from the minimum gap.

Figure 13 shows the results for the roll-separating force coefficient, F(Equation (8b)), which have also been given in Zheng and Tanner [6].The solid line corresponds to the FEM results and the dashed line to

158 E. MITSOULIS

those by LAT. A good agreement is found between the two works, as faras the shape of the curves and the range of values. It is worth noting thatin the present work there is an excellent agreement between the resultsby LAT and FEM, while in the work by Zheng and Tanner [6] there weresome differences especially for lower n.

Figure 14 presents the results for the power coefficient, E (Equation(10b)), obtained from LAT and FEM. It is seen that LAT predicts highervalues than FEM. The discrepancies are due to the differences in thevalues of shear stress on the roll surface, because in FEM there is noapproximation for the arc of a circle as in LAT. These discrepanciesincrease with a decrease in the power-law index (differences in the orderof 14% for n¼ 0.2).

–120 –100 –80 –60 –40 –20 0

–120 –100 –80 –60 –40 –20 0

–120 –100 –80 –60 –40 –20 0

x/H0

–30

–20

–10

0

10

20

30

y/H

0

STR1.000.830.670.500.340.170.01

–0.16–0.32–0.49–0.65

x/H0

–30

–20

–10

0

10

20

30

y/H

0

U1.271.130.990.850.710.570.430.290.150.01

–0.13

x/H0

–30

–20

–10

0

10

20

30

y/H

0

V0.600.480.360.120.020.00

–0.02–0.12–0.36–0.48–0.60

Calendering-1,this workZ&T(a)

(b)

(c)

Figure 11. Comparison between results for calendering-1 of power-law fluids withn¼ 0.4, by Zheng and Tanner [6] (Z&T) and of this work. Contours of flow variables: (a)stream function, (STR), (b) axial velocity u (U), and (c) transverse velocity v (V). Thestream function has been normalized by the flow rate Q, while the velocities have beenmade dimensionless by the roll speed.

Detachment Point in Calendering of Plastic Sheets 159

–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.0 0.5 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

n=1n=0.8n=0.6n=0.4n=0.2

Pre

ssur

e, P

/ P

new

t, m

ax

Dimensionless distance, x ′ = x / (2RH0)1/2

Power-law fluids

Figure 12. Dimensionless pressure distribution for power-law fluids obtained from FEMin calendering-1.

0.0 0.2 0.4 0.6 0.8 1.0

F

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

LATFEM

Power-law fluids

Power-law index, n

Figure 13. Roll-separating force coefficient, F, for power-law fluids obtained from LATand FEM in calendering-1.

160 E. MITSOULIS

Kistler’s Method

Here we revisit the method for finding the detachment point within aFEM context and refer to the pioneering work by Kistler [13], who in hisPhD thesis set the framework for accurately analyzing coating flowswith static and dynamic contact lines via FEM. In calendering, thepoints of attachment and detachment are dynamic contact lines (in a 3D

0.0 0.2 0.4 0.6 0.8 1.0 1.2

E

2

3

4

5

6

7

8

LATFEM

Power-law fluids

Power-law index, n

Figure 14. Power coefficient, E, for power-law fluids obtained from LAT and FEM incalendering-1.

1

2n1

VC,1

VC,2C

d

Dhn2

Figure 15. Details of velocity field in FEM formulation with double-valued nodes for thevelocity at the dynamic contact point, C.

Detachment Point in Calendering of Plastic Sheets 161

sense), since there the velocity changes from a nonzero set value (the rollspeed) to that of the free surface, which is unknown a priori. Theproblem of also locating the unknown contact point (in a 2D sense) issolved by an extra equation setting the contact angle (a known valuefrom experiments or a parameter in the simulations). However, thisholds for coating liquids with non-negligible values of surface tension,which give rise to finite values of the capillary number, Ca. The contact-angle method has been excellently exploited by Scriven’s group [14–17].

However, for very viscous fluids with negligible surface tension(Ca!1), such as polymer melts used in calendering, the above methoddoes not hold. Kistler and Scriven [14] state that ‘the apparent dynamiccontact angle is determined solely by hydrodynamic forces and becomesa dependent variable’. In sample calculations of curtain coatingflows, they show that the dynamic contact angle D ranges between1608 and 1808.

The details of such a determination by hydrodynamic forces are nottrivial and are only given in Kistler’s thesis [13]. Here, we reproduce thismethod (adapted for calendering) and show the difficulties in imple-menting such a scheme in a FEM program.

With respect to Figure 15, we consider the two elements 1 and 2,sharing the detachment or contact point, C, the location of which isunknown. The method consists of using double nodes at the interfacebetween the two elements, which will take different values for thevelocity and the pressure. In the formulation with double-valued velocityat the dynamic contact point, velocity and pressure are discontinuousalong the spine (here a vertical line, ¼ 908) through the contact pointover the length of one element. In element 1, the requirement of nopenetration at the roll surface, �n � �vC,1 ¼ 0, is set as an essentialcondition. The tangential velocity at the roll, �t � �vC,1, is a predictedvariable. On the other hand, velocity �vC,2 at the contact point in element2 is part of the computed prediction; it comes close to being tangential tothe free surface by virtue of the kinematic boundary condition, �n � �v ¼ 0.

Separate momentum residuals on either side of the discontinuityweighted with the shape functions, �i,s , belonging to the nodes ofdiscontinuity, furnish the additional algebraic equations to cover all theunknown velocity coefficients. The momentum conservation principlerequires continuity of momentum flux �n � ��� across the line ofdiscontinuity. In the weighted boundary residual contribution thatarises for element 2, �n2 � ���2 is substituted by � �n1 � ���1, which is computedfrom the deformation field in element 1. The boundary residualcontribution that arises for element 1 is substituted in analogousfashion. The mass conservation principle requires continuity of mass

162 E. MITSOULIS

flux �n � �v across the line of discontinuity. In the present formulation thecondition, Equation (13),Z

�h

�n � ð�v1 � �v2Þds ¼ 0, ð13Þ

furnishes an algebraic equation to cover the additional pressureunknown at the contact point.

Note that the same procedure must be employed for the attachmentpoint E of Figure 2.

The formulation with double-valued velocity introduces two addi-tional, arbitrary parameters: , the angle of inclination of the spinethrough the contact point (in calendering this is set to 908); and �h, thelength over which the flow field is discontinuous along that spine. Unless�h! 0, the computed predictions are sensitive to the two parameters.Kistler [13] shows how these parameters influence the results givingvalues of the dynamic contact angle in the range of 16085D51808, andthese values get closer to 1808 as the Reynolds number, Re, decreases[14]. For polymer melts for which Re551, the approximation of 1808 is,therefore, a very good approximation (but still an approximation for thepurists).

In the course of the present work, many efforts were made toimplement the above method, drawing from our past experience withdouble nodes [18]. However, in Mitsoulis [18] double nodes are usedalong all the elements on an interface and allow only for discontinuouspressures but equal velocities (a requirement for interfaces in multilayerflows). The present requirement of having only one element with doublenodes and discontinuous velocities and pressures upsets the automaticnumbering of the mesh and the corresponding degrees of freedom, andneeds extra programming in the frontal technique of the solution, whichproved too hard to implement.

Thus, it still remains an unresolved problem to implement Kistler’smethod and exactly determine the detachment (and attachment) pointsof sheets in calendering using a u-v-p-h formulation with finite elements.

CONCLUSIONS

The work presented above confirms the utility of lubrication analysisfor power-law fluids, and the validity of the Swift exit boundaryconditions (P¼ dP/dx¼ 0 at x0 ¼ �) in these cases. The lubricationapproximation has been used to derive numerical solutions for the

Detachment Point in Calendering of Plastic Sheets 163

dimensionless leave-off distance, �, and sheet thickness in calenderingfinite sheets of power-law fluids. The 2D FEM analysis was then used tosolve for the flow field including upstream and downstream freesurfaces. It was found that shear-thinning decreases the big vorticespresent in the fluid bank and eliminates them for extremely shear-thinning fluids (n� 0.2). Swelling at the exit is almost nonexistent, withvalues around 1% and decreasing as shear-thinning increases. Theoperating variables (maximum pressure, roll-separating force, andtorque) are as predicted by LAT.

A major effort was made to determine exactly the detachment point byusing Kistler’s method of double nodes for discontinuous velocities andpressures at the dynamic contact point. This method furnishes the extraconditions for locating the detachment point without setting the contactangle, which is part of the solution. However, its implementation in au-v-p-h formulation with finite elements was found not to be a trivialmatter. Thus, the exact location of the detachment (and attachment)points in calendering remains an unresolved issue with FEM.

FUNDING

This work was supported by NTUA [grant number PEBE 2009–2011].

REFERENCES

1. Middleman, S. (1977). Fundamentals of Polymer Processing, McGraw-Hill,New York.

2. Ardichvili, G. (1938). Kautschuk, 14(1): 23, ibid 14: 41.

3. Sofou, S. and Mitsoulis, E. (2004). Calendering of Pseudoplastic andViscoplastic Sheets of Finite Thickness, J. Plastic Film & Sheeting, 20(4):185–222.

4. Mitsoulis, E., Vlachopoulos, J. and Mirza, F.A. (1985). Calendering Analysiswithout the Lubrication Approximation, Polym. Eng. Sci., 25(1): 6–18.

5. Agassant, J.-F. and Espy, M. (1985). Theoretical and Experimental Study of theMolten Polymer Flow in the Calender Bank, Polym. Eng. Sci., 25(2): 113–121.

6. Zheng, R. and Tanner, R.I. (1988). A Numerical Analysis of Calendering,J. Non-Newtonian Fluid Mech., 28(2):149–170.

7. Mitsoulis, E. (2008). Numerical Simulation of Calendering ViscoplasticFluids, J. Non-Newtonian Fluid Mech., 154(2–3): 77–88.

8. Luther, S. and Mewes, D. (2004). Three-dimensional Polymer Flow in theCalender Bank, Polym. Eng. Sci., 44(9): 1642–1647.

164 E. MITSOULIS

9. Hannachi, A. and Mitsoulis, E. (1993). Sheet Coextrusion of PolymerSolutions and Melts: Comparison Between Simulation and Experiments,Adv. Polym. Technol., 12(3): 217–231.

10. Mitsoulis, E. (2005). Multilayer Film Coextrusion of Polymer Melts:Analysis of Industrial Lines with the Finite Element Method, J. Polym.Eng., 25(5): 393–410.

11. Abdali, S.S., Mitsoulis, E. and Markatos, N.C. (1992). Entry and Exit Flowsof Bingham Fluids, J. Rheol., 36(2): 389–407.

12. Mitsoulis, E. (2007). Annular Extrudate Swell of Pseudoplastic andViscoplastic Fluids, J. Non-Newtonian Fluid Mech., 141(2–3): 138–147.

13. Kistler, S.F. (1984). The Fluid Mechanics of Curtain Coating and RelatedViscous Free Surface Flows with Contact Lines, PhD Dissertation,University of Minnesota, MN, USA.

14. Kistler, S.F. and Scriven, L.E. (1984). Coating Flow Theory by FiniteElement and Asymptotic Analysis of the Navier-Stokes System, Int. J. Num.Meth. Fluids, 4(3): 207–229.

15. Coyle, D.J., Macosko, C.W. and Scriven, L.E. (1986). Film-splitting Flows ofForward Roll Coating, J. Fluid Mech., 171(1): 183–207.

16. Coyle, D.J., Macosko, C.W. and Scriven, L.E. (1987). Film-splitting Flows ofShear-thinning Liquids in Forward Roll Coating, AIChE J., 33(5): 741–746.

17. Coyle, D.J., Macosko, C.W. and Scriven, L.E. (1990). The Fluid Dynamics ofReverse Roll Coating, AIChE J., 36(2): 161–174.

18. Mitsoulis, E. (1987). Use of Double Nodes in the Numerical Simulationof Multilayered Flows of Immiscible Fluids, Commun. Appl. Num. Meth.,3(1): 71–76.

BIOGRAPHY

Evan Mitsoulis

Evan Mitsoulis received his Diploma in Chem. Eng. from the NationalTechnical University of Athens (NTUA), Greece, before obtaining hisM.Sc.E. from the University of New Brunswick (Canada) and then hisPhD in ChE from the McMaster University (Canada). From 1984–2000he was a Professor in the Department of Chemical Engineering at theUniversity of Ottawa, Canada, and since 1998 he is with the School ofMining Engineering and Metallurgy at NTUA, where he is the Directorof CAMP-R&D (Computer-Aided Materials Processing – Rheology &Design) Laboratory. He is the author of over 130 refereed papers andover 230 presentations in Conferences on Polymer Processing, Rheology,and Design.

Detachment Point in Calendering of Plastic Sheets 165