Calendering

8
Calendering
■■8.1 Introduction
Calendering covers two distinct types of processes:
ƒƒ Film calendering, for the production of thick sheets (from 300 µm to a few millimeters), typically for PVC or elastomeric compounds. Most calendering machines
contain more than two rolls (up to five in some cases) with a controlled temperature close to the temperature of the polymer. This process is characterized by the
existence of a large “bank” of material upstream from the gap between each pair
of rolls.
ƒƒ Postextrusion calendering used to improve the thickness uniformity and to cool
down a polymer sheet or film downstream of an extrusion line. These calendering
machines have two and sometimes three cooling rolls. The “bank” of material
upstream from the gap between the rolls is much smaller, or even invisible to
the naked eye. This finishing calendering process may be applied to any type of
thermoplastic polymer.
Remark: This second type of calendering process is not to be confused with polymer
coating on a substrate (such as metal or paper). The external appearance of the
machine looks identical, but one of the two rolls can be deformed under pressure
(a metal core surrounded by a rubber band), and it is the deformation of this roll
that ensures intimate contact between the polymer and the substrate (Sollogoub
et al., 2008).
We present below the two types of calendering processes.
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8 Calendering
■■8.2 Rigid Film Calendering Process
8.2.1 Presentation
Calendering was developed in the early 20th century to make rubber sheets used
for the production of treads for tires. This process is still used today for elastomers,
but it was considerably developed in the mid-20th century for the manufacture of
packaging (rigid PVC), oilcloth, or imitation leather (plasticized PVC). Currently,
calendering is mostly used for flooring applications (plasticized PVC reinforced with
high loadings of calcium carbonate, for example). Nowadays, PVC compounds are
sometimes replaced by polyolefin compounds, but it is still marginal. In contrast with
extrusion or injection-molding machines, where all the transformations occur in the
same machine, the calendering process consists of successive units that each play
a specific role. A calendering line is much more expensive than an extrusion line
(up to a factor of 20 to produce sheets of equivalent dimensions), but the production
rates are much larger (up to 6 tons per hour for a 4 m width calendering line with
plasticized PVC). Calendering is preferred to extrusion in the case of PVC because
the risk of damaging the equipment due to polymer degradation is limited as the
flow is not confined in a calender, as is the case in an extruder. In the following, the
rigid PVC calendering process (that is to say without plasticizer) is presented. The
successive units and processes are described in Figure 8.1.
In phase M (mixing), the compound is obtained by dry mixing of the PVC resin
(pellets or powder) and various additives (stabilizers whose function is to inhibit the
degradation reaction; internal and external lubricants, which respectively reduce
the frictional forces occurring between the PVC powder grains and that between the
PVC powder grains and metal surfaces; and processing aids to improve PVC gelation,
melting, fluidity, and processing performance). The formulation of PVC compounds
is generally a closely guarded secret.
Figure 8.1 Calendering line: M: mixing of additives to PVC, G: gelation, H: homogenization,
F: filtration, A: feed, C: calendering, E: extraction, R: cooling, En: windup
8.2 Rigid Film Calendering Process
In phases G (gelation or plastication) and H (homogenization), the compound is
kneaded between heated rolls rotating in opposite directions, in order to transfer
energy very gradually between the grains of the PVC powder to obtain a gel without
degrading the polymer (in the case of PVC, gelation is used instead of melting because
the grain structure disappears completely at temperatures above which the PVC will
degrade). There are generally several sets of kneading rolls, and PVC sheets are cut
and transferred to the next set of kneading roll by a conveyor belt.
Phase F (filtration) corresponds to the passage of the plasticated compound through
a very short extruder containing a filter to retain impurities or insufficiently gelled
PVC particles that could affect the product quality or deteriorate the rolls.
Phase A (feed) consists of a conveyor belt that will transfer the molten PVC extrudate
to feed the bank between the first calender rolls by an alternating movement along
its entire width.
Phase C is the actual calendering step. The calender consists of three, four, or five
rolls, arranged in various configurations (reverse L as illustrated in Figure 8.1, or
in S, Z, or W configurations as discussed by Agassant and Hinault, 2001). The steel
or cast rolls of diameters up to 1 m and width up to 4 m are heated. Their surface is
chrome treated and mirror polished to give a good surface aspect to the PVC sheet.
Typically, each roll has an independent drive. Due to the dimensions of the rolls and
the high viscosity of PVC formulations, the power of the electric motors is important
(100 to 200 kW for rolls of 800 mm diameter and 2 m width). The temperature of
the roll heating system varies between the first and last roll (but still close to the
temperature of the polymer). The rotational speed of the rolls can vary as well, and
the differential peripheral velocity between two successive rolls is called “friction,”
typically between 5 and 30%, but it may be much more important for highly loaded
formulations or polyolefins.
Phase E is the extraction of the sheet, consisting of a series of rolls of small diameter
with a small draw ratio between two successive rolls.
Phase R is the cooling carried out on cooled rolls of larger diameter.
Finally, in phase En, the PVC sheet is wound on a mandrel. The problem is avoiding
localized thickness variations that develop progressively, creating rope-like defects
on the wound-up sheet.
8.2.2 Calendering Problems
There are several problems that one can encounter when producing polymer sheets
by calendering.
ƒƒ The sticking of the sheet on the last roll of the calender (point C of Figure 8.1)
could make it difficult to extract. In the same manner, the sheet may stick to the
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upper roll instead of being driven by the lower one at the gap exit. To remedy this
problem, one can add an external lubricant to the compound in phase M, but if in
too-large quantity this lubricant can affect the flow of the polymer between the
calender rolls.
ƒƒ The deformation of the roll caused by the pressure generated by the calender
results in a greater film thickness in its center than at its periphery, as illustrated in Figure 8.2(a). This thickness is controlled continuously downstream of
the calender by a  radiation gauge, which moves periodically in the transverse
direction of the sheet and, thus, continuously reads the sheet thickness profile.
The use of curved rolls with a diameter at its center slightly larger (by several
hundred micrometers) than that at its periphery can correct this thickness variation. This technique is, however, very restrictive since it would theoretically lead to
designing a roll geometry for each PVC formulation and each calendering condition
(roll speed and gap between the rolls). More versatile techniques are generally
preferred; they either consist in applying a variable constraint outside the roll
bearings (called roll bending) (Figure 8.2(b)) or in slightly misaligning the axes of
rotation of the two rolls (Figure 8.2(c)). These two techniques will reduce the gap
between the two cylinders at the center and increase it at the periphery. In some
cases, curved rolls, roll bending, and roll axis misalignment may be combined.
The forces generated by the calendering could deform the frame of the calender,
and the produced sheet could be thicker than the initially set gap between the rolls
before processing the material. Finally, the calendered sheet could exhibit defects
(matteness, chevrons, air bubbles) of more or less serious importance, depending
on the final use of the product. These defects will be discussed in Section 10.3.
bearing
F
F
F
(a)
F
(b)
(c)
Figure 8.2 (a) Elastic deformation of the rolls; (b) correction of the elastic deformation of the roll
by applying a constraint outside the bearings (roll bending); (c) misalignment of the rolls (top view)
8.2 Rigid Film Calendering Process
8.2.3 Aim of Calendering Process Modeling
Through the modeling of the calendering process one can predict the force and torque
exerted on the rolls. This makes it possible to design a calendering unit and to calculate the size of the rolls and the power required of the driving motors for a range of
PVC compounds with known rheological properties under operating conditions that
are economically attractive. Modeling also allows us to assess a priori the importance
of roll bending or of misalignment of the rolls to obtain a sheet of uniform thickness
and so to save time for a new production. Modeling can also predict how important
is the viscous heating of the material, which is a key problem in PVC processing.
This allows us to adjust accordingly the temperature and the rotational speed of the
rolls to operate below the degradation conditions. Finally, modeling is used to relate
the appearance of defects to critical values of thermomechanical parameters (stress,
pressure, for example) and, hence, to apprehend the phenomena and then delay the
onset of defects. This point will be discussed in Section 10.3.
8.2.4 Kinematics of Calendering
In the following we focus on the flow in the gap between the last two rolls of the
calender. It is at this level that the final quality of the sheet (size, surface appearance)
is determined, and, therefore, process control requires a good understanding of this
stage of the process. The flow is very complex, as shown in Figure 8.3.
Figure 8.3 A schematic representation of the flow kinematics in a calender bank:
(a) cross section; (b) 3D flow lines for which half a bank is represented (from Unkruer, 1972)
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The flow proceeds as follows:
1. The sheet in close contact with roll  (the lower roll in Figure 8.3) is fed as illustrated in sketch (a) of the figure.
2. A part of the material, of thickness hAl, goes directly through the nip along roll .
3. Another part of the material, of thickness hE – hAl, is recirculated to form a bank.
4. A portion of the bank material (hA2) will flow in the nip along roll  (upper roll
in Figure 8.3).
5. The rest of the material (hE – hAl – hA2) flows in the transverse direction, thus
increasing the width of the sheet.
A simplified approach is first proposed, based on the hydrodynamic lubrication
approximations (Figure 8.4). It is assumed that the flow is purely two-dimensional,
that is to say that the transverse flow along the axis of the rolls of the calender
(Figure 8.3(b)) is negligible. It is also assumed there is no slippage between the
polymer and the rolls.
y
R
H
h
h*
B
A
C
U
h0
x
D
Pressure
E
F
x
Figure 8.4 Simplified flow kinematics between the rolls of a calender and the resulting
pressure profile (from Agassant and Hinault, 2000)
This analysis is based on the analogy between the flow in the gap of the rolls of a calender and that of a lubricated roller bearing. The analysis presented in Section 4.5.2.3
reveals the existence of a pressure peak between the rolls, which is located upstream
of the roll nip, and a final sheet thickness that is larger than the gap between the
rolls. This is called the spread height or the thickness recovery, r = h * h0 (2h0 is
the roll gap, and 2h* represents the thickness at point C of Figure 8.4 where the
pressure is maximum; it is also equal to the thickness of the sheet at the exit).
8.2 Rigid Film Calendering Process
In the case of a speed differential (so-called friction) between the rolls (the speed of
the lower roll is greater than that of the upper one), in particular for increasing the
viscous dissipation and improving the PVC gelation, it is assumed that there is no
spread height and that the sheet leaves the contact with the rolls with the thickness
of the nip (Figure 8.5). Intermediate situations may be observed if the friction is low.
In both cases, the lubrication approximations are valid (see Section 4.4.2.1).
ƒƒ The rolls have diameters typically of the order of 400 mm. The thickness of manufactured sheets is between 200 and 600 µm. Assuming that the roll is a parabola
in the flow area, we can write

x2 
=
h h0 1 +

 2Rh0 
(8.1)
where h(x) is half the thickness in the flow direction (see Figure 8.4). Then:
dh
=
dx
x
=
R
2
(h − h0 )
1
R
(8.2)
since the gap is very small compared with the radius R of the roll.
ƒƒ Similarly, the radius of curvature is the radius of the rolls, which implies
h
<< 1
R
(8.3)
Pressure
0
x
Figure 8.5 Flow kinematics and pressure distribution between the rolls of a calender when
there is a friction between the rolls
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8.2.5 Isothermal Newtonian Model Based on Lubrication Approximations
The first model based on the lubrication approximations was developed in 1938
by Ardichvilli for rubber calendering. It neglected the thickness recovery and,
therefore, assumed that the rubber sheet is leaving the contact with the rolls with
a thickness equal to the gap. Gaskell (1950) and McKelvey (1962) introduced the
concept of spread height, and we present their model below in the case where the
speed of the two rolls is the same.
8.2.5.1 Reynolds Equation
With the lubrication approximations, the Stokes equations are simplified to (see
Section 4.4.2.1)
∂2u
dp
= 2
dx
∂y
(8.4)
and using the boundary conditions shown in Figure 8.4 we obtain a Reynolds equation similar to that obtained in Section 4.5.4:
dp
h − h∗
= 3U
dx
h3
(8.5)
where U is the peripheral speed of the rolls.
Remark: Equation (8.5) expresses that the pressure reaches a maximum at
point C (Figure 8.4; (dp dx) = 0 for h = h∗ ) as proposed in the qualitative analysis
(Section 4.5.2), but also that the pressure gradient is zero at the output contact
point (E).
8.2.5.2 Spread Height Calculation
The spread height is defined as the ratio of the sheet thickness in contact with the
rolls at point E, h*, and the gap between the two rolls (point D), h0. Defining a new
variable a by
a =±
h
x
−1 =
h0
2Rh0
(8.6)
the Reynolds equation may be written as
2Rh0 a2 − a∗2
dp
= 3U
da
h02 (a2 + 1)3
(8.7)
8.2 Rigid Film Calendering Process
with: a∗ =
h∗
− 1=
h0
r −1
(8.8)
− H / h0 − 1 (H is the bank half-thickEquation (8.7) can be integrated from −aH =
ness) to a to obtain, assuming a zero pressure at the entrance,
p(a) = 3U
2Rh0 a
h02
a ′ 2 − a∗ 2
∫−aH (a′
2
+ 1)3
da ′
(8.9)
The expression for a* is obtained by taking the pressure equal to zero at the exit:
a∗
a ′ 2 − a ∗2
∫−aH (a′
2
+ 1)3
da ′ = 0
(8.10)
The spread height as a function of the ratio of bank thickness to gap clearance is
reported in Figure 8.6. As in most operations, the bank is large compared to the gap
(H/h0 > 10), and it is clear from the figure that the spread height is constant, with
a ratio slightly above 1.2 (r = 1.226).
In the case of smaller banks, like those encountered in finishing calendering processes downstream of a sheet extrusion line, the spread height and, hence, the final
thickness of the sheet will be strongly dependent on the bank size.
Figure 8.6 Spread height as a function of the bank-to-nip ratio
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8.2.5.3 Roll Separating Force and Torque Exerted on the Roll
Knowing the value of the spread height, and hence of a*, the pressure profile is
obtained by integrating Eq. (8.9), and the results are presented in Figure 8.7. The
figure shows that the pressure peak is similar to what we had imagined qualitatively
in Section 4.5.2.3 and that it becomes almost independent of the bank thickness as
soon as conventional calendering conditions are used.
Figure 8.7 Pressure profiles for several values of the bank-to-nip ratio
(roll diameter D = 600 mm; gap 2h0 = 0.4 mm; roll speed Ω = 4.8 rpm; viscosity  = 103 Pa·s)
The load or the separating force acting on the rolls can now be calculated by integrating the pressure profile:
 2
∗2 
RUW a∗ a′  a − a 
=
F W=
∫contact p( x) dx 6 h0 −aH −aH (a2 + 1)3 da da ′
∫
∫
(8.11)
where W is the sheet width. Using the equilibrium value for a* deduced from
Figure 8.6 and assuming that aH = ∞ (this is reasonable since the pressure does not
increase as soon as the bank thickness is sufficiently large), the separating force
or load can be written as
F = 1.23  RUW
h0
(8.12)
Similarly, the calendering torque is expressed by
C =W∫
contact
R t w ( x)dx
(8.13)
where t w is the shear stress at the cylinder wall, which can be expressed by the
Stokes equation as t w = h dp dx. Finally, the expression for the torque is
8.2 Rigid Film Calendering Process
C = 1.62 RUW
2R
h0
(8.14)
Equations (8.13) and (8.14) are of special interest to the design engineer as they
allow for the calculation of
ƒƒ the elastic deformation of the rolls, and hence of the curvature of the plastic sheets
produced by calendering; and
ƒƒ the power required to rotate the calendering unit.
Using the data on which Figure 8.7 is based, we get the following numerical values
for a calendering width W equal to 1 m:
ƒƒ Separating force or load: F = 28 × 104 N
ƒƒ Resisting torque:
C = 4 × 103 N·m
These values greatly overstate the reality for the usual calendering conditions like
those proposed in Figure 8.7. This leads us to question the kinematic assumptions
used and the Newtonian behavior.
8.2.6 More General Newtonian Models
8.2.6.1 Two-Dimensional Model
The assumption that the flow in the bank as illustrated in Figure 8.3 can be made
symmetrical as done in Figure 8.4 is somewhat unrealistic. Removing this assumption, but considering a two-dimensional flow (i.e., neglecting the transverse flow),
the Navier-Stokes equations for this problem have been solved in terms of the stream
and vorticity functions by Agassant and Espy (1985) (see Appendix 1, Section 8.4.1)
and in terms of velocity and pressure by Mitsoulis et al. (1985).
The calculated streamlines in the bank are very close to those observed experimentally by stopping the rolls, sampling in the bank, cooling and cutting it into thin
slices, and then polishing the solid samples. The photograph in Figure 8.9 illustrates
the experimental streamlines compared to the computed ones of Figure 8.8.
Figure 8.10 shows that the pressure profile calculated on the axis of symmetry
between the rolls is quite similar to that obtained under the lubrication approximations, which validates the simplified approach presented above.
Remark: A three-dimensional calculation of the flow in the calendering bank,
thus taking into account the cross-flow along the axis of the rolls (Figure 8.3),
has been proposed by Luther and Meves (2004). It predicts the sheet enlargement
as it goes through the nip of the rolls, but the pressure peak between the rolls is
only slightly modified.
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Figure 8.8 Two-dimensional computed streamlines in the calendering bank
(from Agassant and Espy, 1985)
Figure 8.9 Visualization of the flow in a section of the polymer bank
Figure 8.10 Pressure profile on the axis of symmetry:
(···) finite elements method, () lubrication approximations; diameter of the rolls D = 225 mm;
rotation velocity Ω = 9.5 rpm; gap 2h0 = 0.245 mm; viscosity  = 103 Pa·s
8.2 Rigid Film Calendering Process
8.2.6.2 Influence of Slippage between the Polymer and the Rolls
PVC compounds can possibly slip on the calender rolls, especially when the shear
stress at the roll surface is large. Given the kinematic analysis of Figure 8.4, it is
speculated that above a fixed threshold stress, the velocity of the polymer along the
rolls will be less than the peripheral velocity of the rolls upstream of the point of
maximum pressure (the shear stress is positive); on the contrary, downstream of the
point of maximum pressure, the velocity of the polymer along the rolls, above the
same threshold stress, will be greater than the linear velocity of the rolls (the shear
stress is negative). As shown in Figure 8.11, at the point of maximum pressure and
in its vicinity (between points K and G), there is no sliding since the shear rate (and
thus the shear stress that is proportional to the shear rate in the case of a Newtonian
behavior) is low or even zero at point C. The same situation is encountered in the
vicinity of the flow outlet (between points H and E). For both areas, the Reynolds
equation (Eq. (8.5)) is expressed as in the original analysis of Section 8.2.5.1.
y
v
u
v u
v
h*
A
B
C
K
v
G
v
v u
D
x
H
E
Figure 8.11 Sketch of the flow between the rolls of a calender with sliding
Conversely, upstream from point K and between points G and H, the Reynolds equation needs to be rewritten considering a slip velocity (Agassant, 1980):
dp
h − h*
u( x)
= 3U
+ 3 2
3
dx
h
h
(8.15)
The sliding (or slip) velocity, u(x), (i.e., the difference between the velocity of the
roll and the velocity of the polymer in contact with the roll) is negative upstream
of point K and positive between G and H. Equation (8.15) shows that slippage will
cause a decrease in the pressure gradient (in absolute value) for both the positive
and negative sliding areas and, therefore, a pressure decrease in the calender nip.
Sliding areas are determined in writing that the slip velocity vanishes as soon as
the shear rate is lower (in absolute value) than a critical value, which is a parameter
of the model. The slip velocity is then determined at each point of the gap by using
the continuity of the flow rate. As a consequence, this model does not require introducing a slip expression. The pressure predictions depend only on the value of the
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Figure 8.12 Influence of the critical shear rate on the normalized pressure distribution in a
calender gap (from Agassant, 1980)
critical shear rate at which slippage appears. Figure 8.12 shows that the pressure
decreases significantly when the value of the critical shear rate decreases and the
polymer is allowed to slide over larger areas. The same result has been obtained by
Ray and Shenoy (1985).
Furthermore, the value of the film thickness at the gap exit (the spread height as
defined in Section 8.2.5.2) and the flow rate increase with slippage (r = 1.226 for
a sticking contact and r = 1.271 for a perfectly sliding contact). This result is not
intuitive as the output rate results from the contributions of the pressure flow
(which increases with pressure) and of the drag flow by the roll (which increases
because the polymer velocity is larger than that for the sticking case after the point
of maximum pressure). In this case, the mechanism of slippage is dominant.
8.2.6.3 Calendering Analysis When Introducing a Velocity Differential
between the Rolls
A difference of velocity between the two rolls, (U1 and U2), is generally referred to
as calendering with a friction coefficient f = U2 U1 . The preceding models need to
be enriched (Ehrman et al., 1977; Ramli Wan Daud, 1986; Magnier and Agassant,
2013). A very similar Reynolds equation is obtained, written as
dp 3
h − h*
=
(U1 + U2 ) 3
dx 2
h
(8.16)
This equation can be integrated assuming that the polymer film leaves the upper
roll at the nip, as explained in Figure 8.5. The pressure is lower than if assuming a
U (U1 + U2 ) 2
calendering process with a roll velocity equal to the mean velocity =
and the existence of a spread height (Figure 8.13).
8.2 Rigid Film Calendering Process
Figure 8.13 Pressure profile in the calender gap with and without accounting for the spread
height: H/h0 = 10,  = 103 Pa·s, h0 = 0.4 mm; () average velocity analysis: U = 150 mm·s–1;
(- - -) case with friction: U1 = 100 mm·s–1, U2 = 200 mm·s–1.
Nevertheless, the roll-separating force values remain of the same order of magnitude
as those previously calculated.
8.2.6.4 Conclusions of the Different Newtonian Models
The pressure profile is of the same order of magnitude when accounting for the
actual kinematics in the calender bank or applying the lubrication approximation.
Similarly, the friction between the rolls and its impact on the exit point of the
polymer sheet does not change the magnitude of the roll-separating force. However,
the introduction of a nonsticking contact between the polymer and the rolls has a
significant influence.
8.2.7 Shear-Thinning Calendering Model
The shear rate varies significantly in the flow domain between the rolls, which
requires considering the shear-thinning behavior of the polymer. Models for the
calendering of shear-thinning polymers have been developed by McKelvey (1962),
Pearson (1966), Chong (1968), Brazinsky et al. (1970), and Agassant and Avenas
(1977), all using the power-law expression for the viscosity. Alston and Astill (1973)
have made use of a hyperbolic-tangent fitting model. Kiparissides and Vlachopoulos
(1976) have compared the results obtained by a finite element method for the two
rheological models. The results obtained with a power-law model will be presented
below.
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8.2.7.1 Generalized Reynolds Equation
Using the lubrication approximations, the dynamic equilibrium equations are
written as
n −1
∂  ∂u ∂u 
dp


=K
∂y  ∂y
∂y 
dx


(8.17)
Equation (8.17) must be integrated in two parts, since the sign of the velocity gradient changes with the axial position (see Figure 8.4):
ƒƒ Downstream of point C: ∂u ∂y < 0 for y > 0
ƒƒ Upstream of point C:
∂u ∂y > 0 for y > 0
Then, downstream of point C, Eq. (8.17) may be written as
dp
∂  ∂u 
− = K
−
dx
∂y  ∂y 
n
(8.18)
and can be integrated with the nonslip conditions on the rolls to obtain the expression for the velocity profile:
u=
n 1

n +1 K
 dp  
 − dx  

1/ n
 (1+ n) / n
 y
− h(1+n)/ n  + U
(8.19)
The expression of the flow rate is
2n  1
Q = W
n +1  K
1/ n
 dp   (1+2n)/ n
+ 2UhW
 − dx   h

(8.20)
In terms of h* (see Figure 8.4), Q = 2WUh* and
n
dp
 2n + 1  (h∗ − h)n
= −K 
U
2n +1
dx
 n
 h
(8.21)
Upstream of point C, the corresponding expression is
n
dp
 2n + 1  (h − h∗ )n
= K
U
2n +1
dx
 n
 h
(8.22)
These two results can be written in the following single equation, which we call the
generalized Reynolds equation:
n −1
∗
n
(h − h∗ )
dp
 2n + 1  h − h
= K
U
dx
h2n +1
 n

(8.23)
8.2 Rigid Film Calendering Process
8.2.7.2 Integrated Generalized Reynolds Equation
We make the same change of variables as done previously and integrate Eq. (8.23)
using the same boundary conditions, that is, the pressure is assumed to be equal to
zero at the inlet and outlet of the contact. The spread height becomes independent
of the bank thickness as soon as the bank is large enough, and as illustrated by
Figure 8.14 the spread height is a weak function of the shear-thinning power-law
index.
On the other hand, Figure 8.15 shows that the pressure profile varies greatly with
the shear-thinning index. It follows that the calendering force and torque are strongly
Figure 8.14 Variation of the spread height with the shear-thinning index n for large banks
Figure 8.15 Pressure profile in the calendering gap as a function of the shear-thinning index n: roll
diameter = 600 mm; rotation velocity Ω = 4.8 rpm; gap 2h0 = 0.4 mm; consistency K = 104 Pa·sn
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Figure 8.16 Variations of the maximum pressure, separating force, and torque with the shearthinning index n (based on the same data as in Figure 8.15)
dependent on the value of n, as reported in Figure 8.16 for the force. As an example,
for a polymer with a K value equal to 104 Pa·sn, we obtain a calendering force equal
to 3 × 106 N if the polymer is Newtonian (n = 1) and only 7 × 104 N for a shear-thinning polymer (n = 0.3, a typical shear-thinning index for a PVC formulation). This
last value for the force is much more realistic.
8.2.8 Thermal Effects in Calendering
The flow of a highly viscous polymer between the rolls of a calendering unit can
generate a considerable viscous dissipation, which cannot be necessarily balanced
by heat conduction to the rolls. This is a major problem in the calendering of PVC,
which is sensitive to degradation. As calculated by Kiparissides and Vlachopoulos
(1978) in a symmetric calendering situation, the dissipated energy is maximum at
the roll wall where the shear rate is maximum, but it is also in the vicinity of the
rolls that the influence of heat conduction will be the largest. A typical development
of temperature profiles is illustrated in Figure 8.17.
The temperature variations are small on the axis of symmetry where the energy dissipation is very low (generated only by the elongation flow as the shear rate is zero if
the two rolls rotate at the same velocity), as can be seen in Figure 8.18. However, the
changes in temperature are much larger in the vicinity of the rolls where the shear
rate is large, and, hence, the viscous dissipation is important. This is illustrated in
Figure 8.18. The temperature first increases and reaches a maximum, then decreases
toward a minimum at the point of maximum pressure as the shear rate is zero, and
the heat transfer is much more effective as it is occurring over a smaller thickness.
8.2 Rigid Film Calendering Process
Figure 8.17 Schematic development of temperature profiles in the gap of a calender for which
the temperature of the roll surface is imposed (from Kiparissides and Vlachopoulos, 1978)
Figure 8.18 Temperature profile along the flow between the calender rolls
(from Kiparissides and Vlachopoulos, 1978)
The temperature increases slightly again in the vicinity of the minimum gap between
the rolls where the shear rate is large and, then, decreases toward the flow outlet
where the shear rate goes to zero.
It is possible to obtain an order of magnitude of the temperature change by calculating
the mean temperature at each axial position within the gap from the bank entrance
to the exit from the contact, by using a slab method as shown in Figure 8.19 (see
Sections 3.3 and 4.6.2). This method may be obviously not valid for the upstream
area of the bank where recirculating flow patterns are observed (see Figures 8.8
and 8.9), but in that flow region the shear rates are small and, therefore, the dissipated power and the temperature change are minimal.
605
606
8 Calendering
Figure 8.19 Solution by slab method for the flow in calendering
Assuming that the polymer has a Newtonian behavior, the thermal energy balance
on a differential volume element 2hWDx yields the following differential equation:
2
h  ∂u 
 T − T0 
dT
rc p hu
=
−Nu k 
 +    dy
dx
 h  0  ∂y 
∫
(8.24)
where u is the mean velocity at x; Nu k (T − T0 ) h is an approximation of the heat
conducted to the rolls, and Nu is the Nusselt number (Section 3.2.5.2). A value of
10 may be deduced from the numerical results obtained by Saillard (1982) for the
shear heating of polymer flow between two parallel plates (Section 3.3).
In fact, when the polymer temperature increases, its viscosity decreases. The velocity
and pressure fields are then changed, and it is necessary to solve successively by an
incremental method the generalized Reynolds equation (Eq. (8.23)) and the thermal
energy balance equation (Eq. (8.24)), taking into account the dependence of the consistency K of the power-law equation as a function of temperature according to the
Arrhenius expression (Section 2.5.1.3) (Agassant and Avenas, 1977; Agassant, 1980):
 = K (T ) g
n −1
(8.25)
Remark 1: For PVC formulations with complex gelation mechanisms, a more
sophisticated temperature dependence for the viscosity may be needed.
Remark 2: It has been supposed that the sheet exit thickness (the spread height)
is not modified by the temperature dependence of the viscosity. Arcos et al. (2011)
found that the spread height decreases when the activation energy of the viscosity
increases.
8.2 Rigid Film Calendering Process
Figure 8.20 Average temperature profile of the polymer in the gap of a calender for several
values of the roll velocity
(diameter D = 550 mm; gap 2h0 = 0.5 mm; consistency K = 104 Pa·sn; power-law index n = 0.3)
Figure 8.20 reports the average temperature profile of the polymer as a function of
its position along the flow direction in the calendering unit for different rotational
velocities of the rolls. The initial temperature of the polymer is supposed to be
equal to the temperature of the rolls (195°C). Within coherence with the pattern of
Figure 8.18, the average temperature rises slowly in the bank region and sharply
in the surroundings of the pressure peak and, then, stabilizes or even decreases
slightly in the final area of the contact where the film thickness decreases and the
Figure 8.21 Pressure profiles in the gap of a calender
(diameter D = 550 mm; gap 2h0 = 0.5 mm; rotational velocity Ω = 5.2 rpm;
initial temperature and roll temperature = 190°C; same rheological parameters as in Figure 8.20);
temperature-dependent model (- - -); isothermal model ().
607
608
8 Calendering
heat conduction is more effective. With increasing roll velocity, the temperature
rise is larger due to the increased viscous dissipation. The temperature drop in the
final zone is less pronounced as the residence time decreases and, thus, conduction
occurs during shorter periods. The calculated increase in the average temperature
between 10°C and 20°C is compatible with the thermal stability of PVC.
The values for the pressure between rolls presented in Figure 8.21 are therefore
somewhat smaller than for the isothermal shear-thinning case.
8.2.9 Viscoelastic Models
Several attempts have been made to account for the viscoelastic behavior in the
calendering process: Agassant (1980) used an upper-convected Maxwell model
and a bipolar coordinate system (Taskerman-Krozer et al., 1975) and showed that
the maximum pressure remains quite identical to the Newtonian case but that the
pressure peak broadens when the Weissenberg number increases, which means
that the roll-separating force increases too. The spread height was not modified.
Zheng and Tanner (1988) found the same kind of result with a Phan-Thien–Tanner
model (Eq. (2.198)) by applying the lubrication approximation (that is, neglecting
the normal stresses in the force balance equations) and using a more sophisticated
2D numerical method. Arcos et al. (2012) and Ali et al. (2015) used the same kind
of constitutive equation and the lubrication approximations and showed that the
spread height was slightly increasing with the Weissenberg number for important
bank dimensions and slightly decreasing for small bank dimensions.
8.2.10 Use of Calendering Models
The separating force F exerted on the rolls may be deduced from the different
pressure calculations that have been presented in the previous sections. It gives us
access to the deformation of each roll along the roll axis, d(z), which will result in
a nonuniform sheet thickness. Assuming that the bearings of the rolls are fixed in
the calender frame, d(z) is expressed as
=
d( z )
F
8 EI
4
4
 W    2z  
−
1

 2 
 W  


(8.26)
where E is the Young’s modulus of the roll, I is its moment of inertia, and W is the
roll length. It is thus possible to define a priori correction methods presented in
Section 8.2.2 (machining of curved rolls, roll bending, or misalignment of the rolls)
in order to obtain a sheet of uniform thickness.
8.2 Rigid Film Calendering Process
Figure 8.22 Roll-separating force as a function of the rotational speed of the rolls:
(●) Newtonian behavior; (■) shear-thinning behavior; (○) temperature-dependent shearthinning behavior (diameter of the rolls: 550 mm; width of the sheet on the rolls: 1.1 m;
gap: 400 µm; roll temperature: 190°C; consistency K = 104 Pa·sn; n = 0.3)
Moreover, this calculation allows us a priori to examine the sensitivity of the process
parameters (rotational velocity and gap, for example) and of the rheological parameters on the roll-separating force and torque.
Figure 8.22 shows that the roll-separating force increases quite linearly with the
rotational velocity of the rolls, but its variations with roll speed are much more
moderate for a shear-thinning behavior and even less when accounting for thermal
phenomena; these results are intuitive.
Figure 8.23 reports the torque on the rolls as a function of the gap between the
rolls. For the processing conditions that have been used here, a Newtonian model
predicts a decrease of the torque exerted on the rolls when the gap increases,
while a shear-thinning and a temperature-dependent shear-thinning model predict
rather modest and larger increases, respectively. These results are nonintuitive.
This reversal of behavior between the Newtonian case and the other more realistic
situations may be different depending on the rotational speed, temperature, and
power-law index.
We will see in Chapter 10 that modeling can also help us to understand the origin
of some calendering defects and optimize the processing parameters in order to
delay the onset of defects.
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610
8 Calendering
Figure 8.23 Variation of the torque exerted on the rolls as a function of the gap:
(●) Newtonian behavior; (■) shear-thinning behavior; (○) temperature-dependent shearthinning behavior (same data as in Figure 8.22, rotational speed of 3.5 rpm)
■■8.3 Postextrusion Calendering Process
8.3.1 Presentation
There are important differences between postextrusion calendering and the traditional calendering process, which was presented in the preceding section. The
rolls are cooled, the bank upstream of the gap is much smaller (two to three times
the gap), and the frame of the calender is much more flexible than in the case of
traditional calendering. A key problem of the postextrusion calendering process is
to adapt the speed of the calender to the output rate of the extruder.
ƒƒ If the extrusion velocity of the sheet delivered by the extruder is larger than the
rotational velocity of the calender rolls, the bank will grow progressively and cause
a widening of the sheet until it will overflow on both sides of the rolls.
ƒƒ If the extrusion velocity of the sheet is smaller than the rotational velocity of the
calender rolls, the bank will disappear, and the calender will not play its role of
improving the thickness homogeneity of the sheet.
As shown in Figure 8.4, the sheet leaves the rolls with a thickness greater than the gap
between the rolls (when the rolls rotate at the same speed), and the thickness recovery
is very sensitive to the bank size when it is small (see Figure 8.6). If the bank thickness
is growing because of an increase of the extruder rate, the thickness recovery will
become more important, and, hence, the flow rate of the calender will increase too.
8.3 Postextrusion Calendering Process
On the other hand, a larger bank will cause an increase in the pressure (Figure 8.7)
and in the separating force between the rolls. As the frame of the calender is flexible,
the gap between the rolls will increase, which will also contribute to increasing the
flow rate of the calender. As a consequence, it is possible to run a sheet-die extruder
and a calender with velocities that are not strictly identical.
Remark: In the traditional calendering process presented in Section 8.2, the bank of
the molten polymer is much more important, and both the thickness recovery and
the force exerted on the rolls are quite insensitive to the bank size. The importance
of this molten polymer reservoir provides a security time for the operator of the
calender to adjust the feed rate of the machine.
8.3.2 Process Modeling
Because of the small size of the bank, the assumption that the flow is symmetrical
with respect to the flow axis is a priori no longer valid. Furthermore, the calender
rolls are cooled at a temperature far below the polymer temperature, which means
that the flow is strongly nonisothermal. Therefore, a 2D finite element model will
be used in the following sections to take into account these aspects.
8.3.2.1 Pressure Field Calculations
A two-dimensional incremental finite element formulation with a Lagrangian description of the flow (Fourment and Chenot, 1994) has been used for the calculations.
It is different from the Eulerian methods presented in Section 4.6.4. It is based on
the commercial software FORGE2© (Serrat et al., 2012). The general approach for
the calculations is as follows:
ƒƒ At the initial time, the extrusion die (left side of Figure 8.24) is filled with polymer,
and a constant flow rate is imposed.
ƒƒ The extrudate falls onto the lower roll and is driven by its rotational speed in the
gap between the lower roll and the upper roll.
ƒƒ A nonslip contact is imposed between the polymer and the rolls by imposing a
very high friction coefficient on the roll surface.
ƒƒ The bank will gradually grow, and, simultaneously, the point where the polymer
film leaves the contact with the upper roll will move downstream, which helps to
increase the throughput.
ƒƒ The calculations are stopped when the shape of the bank and the pressure distribution become time-independent. It is in this sense that this method is called incremental. Note that it is necessary to introduce the gravitational forces in the dynamic
equilibrium equations for the extrudate that falls at the die exit onto the bottom roll
and at the gap outlet for the sheet that leaves the contact with the lower calender roll.
611
612
8 Calendering
Zone B
Asymmetric
cooling
Zone D
Asymmetric
cooling
Q
Die
Zone A
Drawing and
air cooling
Zone C
Calendering
Figure 8.24 2D mapping of the postextrusion calendering process
Figure 8.25 Pressure field in the calender gap in MPa, between 0 (blue) and 0.46 (red):
gap = 0.8 mm; bank size = 2 mm; roll speed = 55 mm·s–1; Newtonian viscosity  = 476 Pa·s.
The figure is presented in the color supplement
This incremental calculation method requires frequent remeshing to reflect the
progressive development of the flow. Figure 8.25 shows that the pressure is uniform
through the gap at each position along the flow when the bank is stabilized, demonstrating a posteriori that the use of lubrication approximations remains valid even in
the case of small banks. It also shows that the sheet leaves the contact between the
rolls well downstream of the nip, which justifies the notion of thickness recovery
introduced at the beginning of the chapter.
8.3.2.2 Temperature Field Calculations
At each time, the mechanical, mass balance, and thermal balance equations
(Eq. (4.90)) are solved in the area occupied by the flow. Then, a new computational
domain is deduced from the velocity field, and the equations are solved in this new
domain and so on until stabilization is reached in the calculation domain for both
the pressure and temperature fields (Figure 8.26). The main problem is to impose
8.3 Postextrusion Calendering Process
appropriate boundary conditions for the heat problem. In Zone A of Figure 8.24
(between the die and the point of contact with the lower roll), a low heat-transfer
coefficient is imposed (around 10 W·m–2·K–1) corresponding to convective heat
transfer (forced or free convection depending on the extrusion velocity); in the B
and D areas, the same heat-transfer coefficient is imposed on the upper part of the
film and a very high heat-transfer coefficient (around 1000 W·m–2·K–1) to the lower
part of the film in contact with the roll (which corresponds approximately to imposing the roll surface temperature). In zone C, a very high heat-transfer coefficient is
imposed on both sides of the film.
Figure 8.26 shows the final temperature distribution: logically the temperature along
the bottom roll is lower than the temperature at the upper roll, which is related to
the fact that the bank is supplied at the lower roll and, thus, with a polymer already
cooled.
Figure 8.27 shows that the temperature profile remains asymmetric with respect
to the flow axis up to the exit of the calender.
Figure 8.26 Temperature field in the calender gap between 66°C (blue) and 220°C (red);
the dimensions of the gap and the bank are the same as in Figure 8.25. The polymer follows
a Carreau viscosity model, the rotational velocity is 33 mm·s–1, the temperature of the rolls is
50°C, and the temperature of the polymer at the die exit is 220°C. The figure is presented in
the color supplement
Figure 8.27 Temperature profile in the film thickness at the calender exit for two different line
speeds
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8 Calendering
This model allows prediction of the required distance at a given roll velocity for a
complete solidification (crystallization) of the film. It will be necessary, in some
cases, to add to the calender unit a third roll to complete the cooling. This model
also predicts the appearance of defects related to the flow in the transverse direction
(along the axis of the rolls). Unlike in traditional calenders where the flow along the
axis of the cylinder originates from the recirculation in the bank, as illustrated in
Figure 8.3, here it is the pressure developed in the gap, coupled with the solidification of the polymer in the calendering direction, that can explain the phenomenon.
■■8.4 Appendix
8.4.1 Appendix 1: Calculations of Two-Dimensional Flow in the
Calender Bank by a Finite Element Method
8.4.1.1 The Stokes Equations in Terms of the Stream and Vorticity Functions
The Stokes equations are classically written in the incompressible case with velocity
components and pressure as unknowns (Section 2.1):
∂p
∂2u ∂2u
= ( 2 + 2 )
∂x
∂x
∂y
(8.27)
∂p
∂2v ∂2v
= ( 2 + 2 )
∂y
∂x
∂y
(8.28)
∂ ∂y and v = − ∂ ∂x . For two-dimensional
The stream function  is defined by u =
steady flows,  represents the path followed by individual particles placed as tracers
in the flow. By definition, the incompressibility equation is satisfied. Equations
(8.27) and (8.28) are written in terms of the stream functions as
 ∂3  ∂3  
∂p
=  2 + 3 
∂x
 ∂x ∂y ∂y 
 ∂3 
∂p
∂3  
=
−  3 +

∂y
∂x ∂y 2 
 ∂x
(8.29)
(8.30)
Eliminating the pressure leads to
 ∂4 
∂ ∂p
∂ ∂p
∂4 
∂4  
( ) − ( ) =  4 + 2 2 2 + 4  =DD =0
∂y ∂x ∂x ∂y
∂x ∂y
∂y 
 ∂x
(8.31)
8.4 Appendix
To avoid solving this equation of high-degree derivatives, the vorticity tensor Ω is
introduced:
1  ∂v ∂u 
1  ∂2  ∂2  
1
W=
−
=
−
+ 2=
− D



2
2  ∂x ∂y 
2  ∂x
2
∂y 
(8.32)
Therefore, solving the Stokes equations (Eqs. (8.27) and (8.28)) is equivalent to
solving
DW = 0
(8.33)
D = −2W
(8.34)
with the stream function  and vorticity tensor Ω as unknowns. The pressure is
obtained from Eqs. (8.29) and (8.30)
Dp =
∂ ∂p
∂ ∂p
( )+ ( )=0
∂x ∂x ∂y ∂y
(8.35)
8.4.1.2 Solving the Stream and Vorticity Equations for the 2D Calendering
Problem (Agassant and Espy, 1985)
The location of the bank free surface is fixed. The boundary conditions on the stream
function are intuitive (Figure 8.28):
y
E
2H
2h*
A
Figure 8.28 Boundary conditions for the stream function
x
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8 Calendering
ƒƒ The roll surfaces are streamlines: one can choose Ψ = −Uh∗ on the upper roll and
Ψ = +Uh∗ on the lower roll.
ƒƒ The surface of the bank is also a streamline; by continuity with the upper roll its
value is Ψ = −Uh∗ .
ƒƒ The velocity of the polymer is uniform at the bank inlet (point A) and at the outlet
of the contact (point E). The stream function will therefore evolve linearly between
−Uh∗ and +Uh∗ .
In contrast, the vorticity is known only where the velocity field is known, that is to
say at the bank entrance (point A) and at the contact exit (point E). Arbitrary conditions must be set on the surface of the rolls and on the free surface of the bank,
and, therefore, an iterative method has been used for solving the system of Eqs.
(8.35) and (8.36). The boundary conditions for the pressure are difficult to define:
ƒƒ Zero pressure is imposed on the surface of the bank and at the contact exit (point
E) as in the classical lubrication models.
ƒƒ The pressure is unknown on the roll surfaces; however, its derivatives can be
related to the derivatives of the vorticity function by
∂p
∂W
= 
and
∂x
∂y
∂p
∂W
= −
∂y
∂x
(8.36)
When the value of the function Ω is known in the whole area, especially along the
rolls, Eq. (8.35) is solved to compute the pressure.
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