7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Numerical simulation of Non isothermal calendering viscoplastic fluids
Jose C. Arcos*, Oscar E. Bautista*, Federico Mendezt and Eric G. Bautista*
IPN, ESIME AZC., SEPI, Av. de las granjas No. 682, col. Santa Catarina Azcapotzalco, Mexico D.F., 02550, Mexico
SUNAM, Facultad de Ingenieria, Ciudad Universitaria, Mexico D.F., 04510, Mexico
jarcos@ipn.mx and obautista@ipn.mx
Keywords: variable viscosity, conjugate heat transfer, HerschelBulkley
Abstract
The heat transfer analysis of Calendering NonNewtonian fluids is an important area on polymer processing. Here, we are
interested in the studying the heat transfer phenomena on calendaring NonNewtonian process using the Rheology model of
HerschelBulkley, taking into account the effects of temperature on the consistency index, also the viscous dissipation. The
general governing equations for viscoplastic behavior are obtained to yielding and unyielding regions. In this study the
important parameters in this analysis are the P6clet and Nahme numbers, the second one relates the temperature gradient
generated by viscous dissipation to the temperature gradient necessary to modify the viscosity. In general form, the pressure
distribution is determined from the momentum equation, and then we can calculate the velocity field, finally the energy
equation is solve to calculate the temperature distribution. The algorithm mentioned above, is developed in an iterative form,
because of the high degree of coupling between the energy equation and de equation of motion. The order of magnitude for
the Peclet and Nahme numbers were 103 and 101, respectively. Finally the influence of power law index and the flow rate on
pressure and temperature are obtained.
Introduction
The heat transfer analysis on Calendering process
is an important area for the design of calendar machines.
The study of this class of the heat transfer problems
represents a fundamental branch that permit to obtain a
better thermal design and control polymer processing
performance. Here, we are interested in studying the
calendaring process, in which the shear stress is
represented by HerschelBulkley model, also the apparent
viscosity of the fluid into this rheology model is variable
on temperature. The process of calendering has been
extensively studied by many researchers over the last 50
years. Starting with Ardichvili (1936), the work was
extended to Newtonian and Bingham plastics by Gaskel
(1950), to power law fluid by McKelvey (1962), Pearson
(1966), to HerschelBulkley fluid by Souzanna Sofou and
Evan Mitsoulis (2004) and recently a numerical simulation
of calendering viscoplastic fluids was reported by Evan
Mitsoulis (2008), and others. NonIsothermal effects have
been studied by Kiparissides and Vlachopoulos (1978),
Ivan Lopez Gomez, Omar Estrada and Tim Osswald
(2006).
Although the foregoing works are essential
contributions to the study of Calendering phenomena, they
were only reserved for those situations where the
momentum equation is not coupled with energy equation,
so the velocity and pressure fields are not affected by
temperature isothermall case). The main contribution in
this study is to investigate the influenced of temperature on
velocity and pressure fields in the limit when de Bingham
tends to zero. In this case the zone of interest is a viscous
fluid that obeys the power law model.
Nomenclature
Geometry parameter
Heat capacity
Distance between rolls (nip)
Inlet thickness
Roll variable curvature length
Consistency index
Thermal conductivity
Power law index
Pressure in physical units
Nondimensionless pressure
Peclet number
Volumetric flow in physical units
Nondimensionless volumetric flow
Characteristic volumetric flow
Roll radius
Modified Reynolds number
Temperature in the fluid in physical units
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
To Temperature at the inlet of the fluid, in physical
units
T Roll temperature
U Roll velocity
iT Axial velocity of the fluid in physical units
u Nondimensionless axial velocity of the fluid in
physical units
jT Transversal velocity of the fluid in physical units
v Nondimensionless transversal velocity of the
fluid
x Axial coordinate in physical units
Y Transversal coordinate in physical units
Y Nondimensionless computational transversal
coordinate
Greek letters
X Nondimensionless axial coordinate
yi Nondimensionless axial coordinate at the inlet
Pf Density of the fluid
r Shear stress
2 Nondimensionless flow rate
71 Nondimensionless transversal coordinate
0 Nondimensionless temperature
Formulation and Order of Magnitud Analysis
The physical model under study is shown in fig. 1. A
calendar having equal sized rolls rotating at the same speed
is shown in fig. 1. At the beginning the temperatures of
the sheet and the roll are To and T respectively. The
radius of the rolls is R, their surface speed is U, and the
separation at the nip is2h The material thickness at the
input is He, and H at the exit of the rolls. Using a scale
analysis, we define the characteristic parameters and the
nondimensional variables. The nondimensional
governing equations of continuity, momentum and energy
for the yielded and unyielded zones are obtained. These
governing equations are coupling with temperature, and we
solved them by finite difference methods. Due to the
symmetry of the physical model, we consider only for
convenience the region that is symmetric with the x axis.
Therefore we select the middle point of the distance
between two rolls as the origin of the coordinate system,
whose y axis points normal to the flow direction and the x
axis points in the longitudinal direction of the flow. An
order of magnitude (see for example Bejan, 1995) is useful
to obtain the nondimensional parameters.
ee e region in (nx)I ; X
T R Roll
U
Fig. 1. Physical model sketch
The general governing equations in physical units are
written in the following form,
a+ a 0
+= 0
ax y
(_8Tt _8^}
p u+v
a x y)
ap 2yx
=p + C
ax ay
_DT D2T DW
pcj = k + 
Dx Sy2
The hydrodynamics and thermal boundary conditions in
physical units for the above equations are set, like
x= Z,
%=Xf,
p = dpdx = 0
p =0, T =T
y = +h(x), i = U, T7 = T
y= 0, =0, =0
From the momentum equation for the fluid in the
longitudinal direction x, it can be shown that the pressure
is of the order of,
pc K(U/ho) (7)
Besides the orders of magnitude of the important variables
on calendaring process are presented in the following
form,
X (2Rh0)1/ y~ ho,
U ~ U,
(9)
Mathematical Formulation
In this section, we present the nondimensional governing
equations needed to solve the calendaring heat transfer
problem. Based on the above order of magnitude analysis,
we introduce the following nondimensional variables.
h(x)
=1 + X2,
h,
/2 = 1,
2Uho
x dP (2Rho)1/2 dp
2Rho d K(U /ho)"dx
=7
ka
u
u= ,
U
Q*= Q
2Uho
(10)
Replacing the nondimensional variables into the
momentum equation eq.(2), we obtain the nondimensional
form of this,
1+z2
Qc 2'7,, (8)
S1/2 U
2R
R u Lu aP 8 unl (11)u
Re u+v = + 
The Reynolds number at eq.(11) is defined as,
Re pfU2 h'
K (12)
Herein, the inertial, pressure and viscous terms are of order
unit, while the Reynolds number is of order 104, in these
processes, hence the left hand side of eq.(11) is negligible
compared with the others one. The relationship (11), can be
reduced as,
aP a( au (u
OX C 7 C9 l J [C97)
If we assume that the two rolls are identical and rotate with
the same linear speed, then appropriate boundary
conditions are,
q7=0, (0
C9r (14)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
P_ (2n+1 (12
2 n ho K+ ( 2)22 (20)
(20)
The pressure field was obtain numerically from eq.(20).
The integration is done by using a fourthorder Runge
Kutta scheme, then eq.(20) is write in the following form,
(X)= 2n+l 2R 1/2 f 22)1 2 X2 n1l
1, ( 1+
(21)
With regard to energy equation, and making a change of
transversal variable to get a rectangular computational
domain, the nondimensional energy equation for the fluids
is,
8 BPe (i+ x22 2 ( n+ 1 r c
(22a)
Replacing the velocity and the viscous dissipation on the
above equation we get,
= 1+X2, u=
The other hand, the pressure boundary conditions are,
1 820
dP
P=0,
dy
) = 0,
0, X=
X=Xf
From the Newtonian solution we can see that we should
expect two flow regions, one where the velocity gradient is
positive and one where the gradient is negative. We shall
have to integrate eq.(13) separately in each region. The
results are
Pe 2n+1" 122 n1
n 1+ 2)2nl yn
(22b)
The boundary conditions for the energy equation (22b) are,
(n )ha d 2 (
Sn+1 2R dZ K
i n 1 ? K j
u n+1 2R d% ['
(1+72)
(1+72) ,'
>0
dZ
= 0= 0
Y=l, 0=0
80
= 0,
8Y
0=0
(22c)
 <0 The equation of energy was solved by a marching finite
difference method of implicit type for the upper half of the
(19) flow field. The following finite difference approximations
were used for the temperature derivatives.
From the continuity equation and assuming that the
velocity profile is flat at the exit of the calendar, we obtain
the nondimensional pressure gradient, then
ao0 Oia1 ,z
OX AX
a200 O+ ,,+1 20, + +1 O 20j, +O ,
8Y2 2AY2
(24)
Replacing the finite difference approximations on eq.(22),
and applying the CrankNicolson method, we obtain the
CrankNicolson finite difference equation for the internal
nodes, defined as,
MJd ,,,1+ +2(+MJd)Oj,,+ M, ,dO =M ,d6
2(1 Mj,d) 0,, +MJ,dO +,, +N,,A
(25)
In the next section, we present the numerical results, on
pressure and temperature fields to calendering
nonnewtonian fluids.
Results and Discussion
The characteristic values of the parameters involved on
calendering process are presented in Table 1
Parameter Symbol Values
Radius R 0.10m
Minimum distance 0. m
ho 0.0001m
between rolls
Power law index n 0.11.0
Roll Velocity U 0. m/s
Dimensionless flow 0.300.60
rate
In the figures 14, the nondimensional axial pressure field
on calendering process is shown for a viscoplastic fluid
obeying the power law. In these figures we can see the
power law index influence on pressure field for various
dimensionless flow rate.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The extrema of the pressure distribution occur at = X,
and z= For a given dimensionless leave of
distance z=2 corresponding to a finite sheets,
integration of the momentum equation is carried out
until the pressure passes through zero At this point
is the entry to the domain and thus the value of y = I ,
is found since here P(z
 )= 0.0
Fig. 2. Axial nondimensional pressure distribution for
different values of the power law index for a viscoplastic fluid
with a dimensionless flow rate / = 0.444
In figure 1, for 2=0.3 and n=l the maximum
pressure is located at = 0.3 where P=4.58, while for the
viscoplastic fluid P=24.11 with a power law index of n=0.2,
since the pressure in the last case is five higher than the
Newtonian case, we can find the influence of the parameter
n, on the pressure field.
In figures 1 and 3, we show the influence of the
nondimensionless flow rate on the nondimensionless
pressure field, for n=0.1 the maximum pressure is shown on
figure 1, where P=24.1115 while the maximum pressure on
figure 3, is 1.56 times.
12 10 08 06 04 02 00 02 04
X
Figure 1. Nondimensional axial pressure field for different
power law index with 1 =0.3
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
40
35
30. n=0.8
25
ff20 
Fig. 3. Axial nondimensional pressure distribution for
different values of the power law index for a viscoplastic fluid
with a dimensionless flow rate 2 = 0.475
On the other hand, for oo the pressure distribution
P 0 this is shown on figure 4. Under this assumption
we get the maximum thickness value of the calendaring
sheet, i.e. when =0.475. From figures 15, we can
see that the pressure into the nip region, exactly on X = 0 is
the half value of the maximum pressure gets during the
calendering process on = 2 .
50
45
40 n=0.4
35
,=O 300
5  =0 600
40 35 30 25 20 15 10 05 00 05 10
Fig. 4. Nondimensional Axial pressure distribution for
different values of nondimensionless flow rate for a
viscoplastic fluid with a power law index n = 0.4
In figure 5, we show the hard influence of the flow rate on
axial dimensionless pressure distribution for a value of the
power law index. Two limits for 2 are shown in these
graphics, for the first case 2/<0.475, it is possible to
determine exactly the point where the rolls bite for the first
time the finite sheet, this point is at, , .
Fig. 5. nondimensional Axial pressure distribution for
different values of nondimensionless flow rate for a
viscoplastic fluid with a power law index n = 0.8
The second limit is when 2 >0.475 because of the
pressure distribution there is a backflow component which
is superimposed onto the drag flow component.
In this case there is a negative flow along the axis and a
circulation pattern develops in this case it was assumed
that the calender was fed with a mass of fluid so large that
an infinite reservoir of fluid existed upstream from the nip,
these can see on figure 5. The development of the
temperature profile is schematically shown in figures
68. The amount of shear is greater near the roll surfaces,
hence the two maxima in the vicinity of these surfaces.
x=0.3
3 o w p=1.0  x=0.258
Pe =700 x=0.216
x=0.174
x=0.132
2 x=0.09
*x=0.048
* x=0.006
* x=0.036
* x=0.078
Sx=0.12
 x=0.162
 x=0.204
x=0.246
x=0.288
00 02 04 y 06 08 10
Figure 6. Transversal Temperature profile for the calendaring
process with a power law index n = 1.0 and Peclet number
Pe = 700 with a nondimensionless flow rate 2I = 0.3 in the
case of Newtonian fluid
In figures 6 and 7, we can the hard influence of the power
law index on the temperature profiles for the calendaring
process, where in the case of Newtonian fluid the
temperature profiles are higher than in the
nonnewtonian case. The increase of temperature due to
the viscous dissipation is more important on the vicinity of
the rolls because of the movement of the rolls, and the
temperature decreases as we approach the center of the nip,
because of the influence of the drag in this zone is
negligible.
n=0.3
x=0.3
=03 x=0.25E
SPe=700 x=0.21
x=0.17
 x=0.132
x=0.09
*x=0.04E
00 0x=0.o00
*x=0.036
*x=0.078
Sx=0.12
x=0.162
0 wh a nns fw x=0.204
x=0.246
.. x=0.288
00 02 04 y 06 08 10
Figure 7. Transversal Temperature profile for the
calendaring process with a power law index n = 0.3 and
Peclet number Pe = 700 with a nondimensionless flow rate
2 = 0.3 in the case of NonNewtonian fluid
In the figures 7 and 8, we can see the influence of the
nondimensionless flow rate on the temperature profile for
a non Newtonian fluid. When the flow rate decreases the
temperature profiles decrease, because of the velocity
profiles decreases too.
)70
)65 n=03
)60 Z~ x=0 27
a55 Pe=700 x=0 213
5o0o x=O 157
)45 x=0043
035 x=0 013
*x=0 07
30 x=0 127
)25 / x=O 184
)20 r x=O 24
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
dissipation can be high enough to be detrimental for such
temperature sensitive materials as PVC and robbers. It is
important to note that the maximum exit temperature at the
exit plane can be substantially smaller than the absolutely
maximum temperature throughout the whole flow field.
In the present study we can see the influence of the power
law index n on pressure and temperature profiles, for
Newtonian and nonNewtonian cases. On the other hand,
we determine the point where the rolls bite the sheet for
some values of the nondimensionless flow rate l. The
pseudoplastic pressure profiles are higher than for the
Newtonian fluid.
Conclusions
In this work we have obtained the pressure and
temperature profiles on calendaring process on a
nonNewtonian fluid. The pressure profiles were
influenced by the power law index. For the
NonNewtonian fluid the thickness sheet at the outlet was
higher than in the Newtonian case. In the upstream of the
pseudoplastic material, the point where the roll bites the
material for the first is before in comparison to Newtonian
fluids. The point of the maximum pressure is located
in = 
Acknowledgements
This work was sponsored by the Instituto Politecnico
Nacional, under project SIPIPN 20090025 and by the
CONACYT, under project 58817.
References
Ardichvili, G. Kautschuk, 14:23,14:41. (1938)
Gaskell, R. E., The calendaring of plastic materials, J. Appl.
Mech., 17334. (1950)
J. M. Mckelvey,. Polymer Processing. John Wiley & Sons, Inc.,
New York.(1962)
J.R.A. Pearson, Mechanical principles of polymer melt
processing, Pergamon Press, Oxford, (1966)
Sofou, S. y Mitsoulis, E., Calendering of seudoplastic and
viscoplastic fluids using the lubricating approximation, J. Polym.
Eng., 24, 505. (2004)
Mitsoulis E., Numerical Simulation of Calendering Viscoplastic
Fluids. J. NonNewtonian Fluid Mech. 154, 7788.(2008)
Kiparissides C., Vlachopoulos J., A study of viscous dissipation in
the calendaring of powerlaw fluids, Polym. Eng. Sci 25, 618.
(1978)
Osswald T. A, Hernandez Ortiz J., Polymer Processing, Hanser,
Munich. (2006)
Figure 8. Transversal Temperature profile for the calendering
process with a power law index n = 0.3 and Peclet number
Pe = 700 with a nondimensionless flow rate 2 = 0.27 in the
case of NonNewtonian fluid
From the previous figures it is apparent that under certain
operating conditions the temperature rise due to viscous
