In this paper we consider thermal effects of polymer flows through a cylindrical die. First, we derive a model for the oscillatory behavior of polymer flow in an extruder given a functional relation between the pressure and flow rate. A simple isothermal but temperature dependent model is constructed to find this relation. Unfortunately, the model is shown to be invalid in the physical regime of interest. We present several arguments to suggest that the isothermal assumption is reasonable but that a more detailed understanding of the small-scale molecular dynamics near the boundary may be required. Second, we show that a simplified model for thermoflow multiplicity in a cooled tube is inconsistent, when the stationary non-Newtonian flow is assumed to be incompressible without radial pressure gradients and without radial velocity. This inconsistency can be removed by allowing for weak compressibility effects in the down-steam area

Berg, J.B.

Bokhove, O.

Peletier, M.

Williams, J.F.

Zwart, H.

English

Mathematics in Industry Information Service Eprints Archive

5Thermal Modeling in Polymer Extrusion
J.B. van den Berg, O. Bokhove, M.A. Peletier, J.F. Williams, H. Zwart
Abstract
In this paper we consider thermal effects of polymer flows through a cylindri-
cal die. First, we derive a model for the oscillatory behavior of polymer flow
in an extruder given a functional relation between the pressure and flow rate.
A simple isothermal but temperature dependent model is constructed to find
this relation. Unfortunately, the model is shown to be invalid in the physical
regime of interest. We present several arguments to suggest that the isothermal
assumption is reasonable but that a more detailed understanding of the small-
scale molecular dynamics near the boundary may be required. Second, we show
that a simplified model for thermoflow multiplicity in a cooled tube is inconsis-
tent, when the stationary non-Newtonian flow is assumed to be incompressible
without radial pressure gradients and without radial velocity. This inconsistency
can be removed by allowing for weak compressibility effects in the down-steam
area.
Keywords
Polymer extrusion, temperature dependence, constitutive relation, compress-
ibility.
5.1 Introduction
A detailed understanding of polymer flow is important in the manufacturing of many
modem plastics. In the extrusion process molten plastic is forced through a small
aperture or "die" by a winding screw. The manufacturer would of course like to get
the highest possible output through such a machine. However, when the flow rate
57
SWI2000 Thennal Modeling in Polymer Extrusion
is too high hysteresis can occur and instabilities in the flow result in defects in the
product (see Figure 5.1).
Different defects are thought to be caused by different physical mechanisms,
such as a stress singularity at the outlet or hysteresis in the flow. This hysteresis
comes from a multivalued relation between flow rate and applied pressure and the
following hysteretic cycle can be observed experimentally (see Figure 5.2). Such
a cycle has been shown to lead to an oscillatory flow regime (Molenaar and Koop-
mans 1994). One model for this behavior is to assume a stick-slip condition at the
polymer-extruder interface while ignoring temperature effects within the polymer
(Den Doe1der, 1999). Alternatively, in this paper we employ a standard no-slip con-
dition at the boundary of the die and allow, in addition, for temperature variation in
the polymer.
Figure 5.1: Surface instabilities in an extruded polymer
IL\{~~1
flow rate
Figure 5.2: Sketch of a hysteretic flow profile
58
Thennal Modeling in Polymer Extrusion SWI 2000
o·
_ flowrateq
Figure 5.3: Geometry of the barrel-die system
5.2 Modeling the barrel-die system
We begin by modeling the physical device not as the screw-driven extruder as used
in manufacturing but rather as two chambers of different radii with a plunger moving
at constant speed forcing the polymer through the aperture. This is reasonable as it
is the exact geometry of experimental rheometers. Because the volume of the barrel
is considerably greater than that of the die, which is entirely open at the far end. We
take the polymer to be compressible in the barrel but not in the die.
The compressibility relates the pressure in the barrel, P, to the density, p, as
p=~~
XP'
(5.1)
where X is the coefficient of compressibility and the dot denotes differentiation with
respect to time, e.g. p = dpl dt, The time rate of change of the mass M in the barrel is
£.1= -q, where q is the flow rate in the die (see Figure 5.3), and the time of change
of the volume V in the barrel is V = vpA where vp and A are the velocity and area
of the plunger, respectively. Defining the barrel density as p = M/ V and with (5.1),
we have
p M V .
p= M --y =XP.
Assuming P = f(q) = fe-M) = J(M), we can simplify (5.2) to
M V - ...
---=Xf'(M)MM V .
(5.2)
(5.3)
It may be seen that (5.3) has the following first integral
- . M-
1na = InM/ V - X f(M) <===> a = - e-xf(M)
V
(5.4)
59
SWI2000 Thennal Modeling in Polymer Extrusion
with integration constant a. Since V = Vo - vp A t we have
V/V = -avpA exp[x](M)]/M. (5.5)
Substitution of (5.5) into (5.3) gives the following autonomous equation
.. a vp A exj(M) +M
M= _.
Mxf'(M)
with a = M(O) exp[-xi(M(O))]/V(O) and l' = dj;dM. The model is closed
when we specify a suitable constitutive relation q = f( - M) = i(M), initial data
and vp, A and X. In the next section, we derive a constitutive relation based on
several simplifying assumptions.
(5.6)
5.3 A temperature dependent constitutive relation
Following Stroh et at. (1990), we consider stationary axisymmetric, incompressible
non-Newtonian flow in the cylindrical die, in which the pressure is independent of
the radial direction and in which the radial velocity is zero. The incompressibility
condition implies that v = vCr) with v the velocity along the die and r the radial
coordinate. The momentum balance in the axial z-direction and the energy equation
take the form (e.g. Stroh et at. 1990)
dp
dz
aT
pCp v az
~~(T}rav),
r ar ar
K a ( aT) (av)2-- r- + T}(T,n) -
r ar ar ar
(5.7)
(5.8)
where T is the temperature, p the pressure, cp the specific heat at constant pressure,
K the thermal conductivity constant, and T}(T, n) the viscosity. The latter is obtained
from experiments and summarized in an Arrhenius relation
I
a
I
n~1v l' -1' av n~1
T}= T}o(T) ar = T}ooeY~ I arl (5.9)
Note that we have also assumed that the problem is steady in time, that is any tem-
perature or pressure changes are assumed to happen instantaneously.
Defining the geometry of the system in Figure 5.4, all the parameters have the
approximate SI or dimensionless values as in Table 5.1 (page 62). Froh et at. (1990)
considered the system (5.7)-(5.8) numerically. Instead, we will derive a constitutive
relation by simplifying this system based on the following assumptions:
• The pressure drop over the die is taken constant.
60
Thermal Modeling in Polymer Extrusion SWI2000
Lz
Die- 2R
L
Figure 5.4: Geometry of the rheometer
• The flow in the die is isothermal which results in a parametric dependence on
the temperature T in the die. The temperature To of the barrel and the die walls
is assumed to quickly change from To to T upon entering the die.
• The heat production by the polymer in the die balances the heat loss through
the die walls.
Hence, we take the pressure gradient to be constant in the die with polymer at a
constant temperature T
dp !l.P
- -
dz L
Using (5.10) and the boundary condition v(R) = 0, we integrate (5.7) from z = 0 to
L and obtain
1 a (Iavl") lavl" s»= -;:ar r ar ==} ar = 21/0 L r
_ n (!l.P)k (RHl ill)
_ - - It -r" .
n + 1 2L1/0
!l.P
1/oL
==} v(r)
(5.10)
(5.11 )
(5.12)
The flow rate q is thus given by
q = 2TCP lR v(r) rdr
1
pTCn ( !l.P )" R3+~. (5.13)
3n+ 1 21/oL
The viscous heat production, the second term on the right in the energy equation (5.8),
is given by
r la 1"+1 (!l.P)"tl
2nL 10 1/0 a~ rdr=2TC110L3n:l 2L110 R3+k. (5.14)
61
SW12000 Thermal Modeling in Polymer Extrusion
Parameter Description Value
n power law coefficient 0.5
1/00 viscosity at To for n = 0.5 l(P-I04 kg/(m.JS)
y Arrhenius coefficient 2.5
To ambient temperature 450K
P pressure in barrel (O.l-lO)MPa
L length of the die 10-2 m
R radius of the die 10-3 m
A cross-sectional area of the barrel 1O-4m2
Vavc average exit velocity (10-3 -10-1) m/s
vp average exit velocity (10-4 - 10-2) m/s
p density of the polymer loJ kg/m!
cp specific heat of the polymer 2 x 103m2/(s2 K)
« thermal conductivity of the polymer 1kgm/(Ks3)
h heat conduction (10 -1Q4)kg/(Ks3)
X compressibility of the polymer 1O-9Pa-1
Table 5.1: Parameter values
Setting the heat production equal to the heat lost through the boundary, i.e.
2rrRLh(T-To)
with h the heat conduction, we have
( )
'!±l ( +1)1tJ.P "=h(T_Tt)3n+l 1/02" " (5.15)
L 0 n R2"+1
Writing tJ.P = P - P. ~ P with P the pressure in the barrel and P. the atmospheric
pressure, we can write (5.13) and (5.15) symbolically as
q=F1(P,T), P = F2(T),
respectively, since 1/0 = 1/00 exp [y (To - T)/ T]. Hence, we have an implicit relation
P = f(q) which is not a cubic, but can still exhibit hysteretic behavior, as we will
show. To investigate the hysteresis, we consider the zeroes and poles of
dP dPdT nP(T2-yTTo/n+yT~/n)
dq = dT dq =q (T2+yToT-yT~) (5.16)
62
Thermal Modeling in Polymer Extrusion SWI2000
The zeroes T± and pole Tp of (5.16) are
T± = ~:o(1 ±JI-4n/y), Tp = Y;O (-1+ Jl +4/Y) (5.17)
respectively. (Note that T > 0.) For the given values of y and n this gives
1
T+/ To = 2: (5 + ../5) '" 3.62,
1
L/ To = 2: (5 - ../5) '" 1.38,
5
Tp/To = 4(-1+.J2."6) "'0.77. (5.18)
We see that there is a bifurcation at y / n = 4 which gives T+ = T_ = 2 To. The pole
Tp < To for every y and there is hysteresis when = y/n > 4, see Figure 5.5. For large
y / n the temperatures take the asymptotic form
T+ y (n)-"'--1+0 - ,
To n y
L '" 1+ 0 (~) .
To y
(5.19)
The relationship between applied pressure and flow rate is shown in Figure 5.6 for
various values of h and 1100.
~6
E::-.
, !..-: _ .
5
y/n
10
Figure 5.5: Bifurcation diagram for temperature dependent polymer flows
63
SWI2000 Thermal Modeling in Polymer Extrusion
a.
2.51 t-;~,I,------~-----
15 1000 2000 3000
T(K)
0.6 i b. i
- 1{~ »>~ 0 5 ,I <, --------------
~. ----~-
... ,
0.4 1000 2000 3000
T(K)
c.
0.251 t>; _.----------.-.
f '-"-'-'-'-'---
~0.2
...
0.15
0.1 1000 2000 3000
T(K)
r2.51/ ~ .i->:- I '--..---------
tj
4000 15 0 4
q (g/s)
62
0.6Ti -------------,
(\
I "-- jl", »>:
~ 05
1
1 .-----_.-......./--
11
4000 0.40 0.5 q (g/s) I 1.5
0.251 r-·"' __._- - ------.---
i
~0.2ll
...
0.15
0.1 04000 0.02 qO(Ws) 0.06 0.08
Figure 5.6: Relation between applied pressure P and flow rate q for a) h =
10"kg/(Ks3) and 1100 = 10"kg/(m,JS), b) h = lOOkg/(Ks3) and 1100 =
5000kg/(m,JS), andc)h= lOkg/(Ks3) and 1100 = l04kg/(m,JS). The
values of the remaining parameters are found in Table 5.1
64
Thermal Mode/ing in Polymer Extrusion SWI2000
5.4 Limitations
The assumptions we have imposed require that we check a posteriori whether the
model is consistent. To that effect, we estimate the temperature difference L\ T =
T - To by equating the work W done in the barrel with the dissipation of heat Q in the
die. Using vp = (10-2 - 10-4) m1s and P = (0.25 - 2.5) M Pa and where necessary
other values from Table 5.1, we find
W=PAvp=Q=rr:R2LpcpL\T <==} L\T=(0.04-40)K, (5.20)
respectively. With an ambient temperature of To = 450K the 140% or 360% temper-
ature change found in our model (see the estimates (5.18» is unfortunately too large.
Alternatively, the Peclet number, the ratio of convection and diffusion, for
pcpR
2
v.vc ~ 0.2 _ 20
Pe= LK (5.21)
for an average die velocity V.vc = (10-3 - 10-1) m1s, respectively. The model as
presented appears valid for the high flow regime, where flow rates are of the order
of grams per second, although the temperature range is clearly almost two orders
of magnitude too high. The magnitude of the heat conduction parameter h is very
poorly estimated, while the results are clearly sensitive to h. At the higher velocities
it is conceivable that there could be a boundary layer with some localized viscous
heating, which analysis could lead to better estimates of h.
5.5 Compressibility effects and consistency
The model (5.7)-(5.8) turns out to be too restrictive, because it is inconsistent to insist
on incompressibility, the absence of a radial velocity as well as a pressure which is
independent of the radial direction. This inconsistency can be shown to arise as
follows. Remember that incompressibility implies v = v(r). First, since we assumed
p = pt.z), (5.7) can be integrated and we find
~oo eF(T) I :~ r p(z)r2 (5.22)
where F(T) = exp Ly (To - T)/ TJ. But at z = 0, we have T = To so
av/ar= er», (5.23)
where C is an integration constant. Hence
~oo eF(T) Cn r = p(z) r
2
(5.24)
65
SWI2000 Thennal Modeling in Polymer Extrusion
and it follows that F(T) and thus is T also a function of z only. If we substitute (5.23)
into (5.8) and also use the previous result that T = T(z), we obtain
et 1 n + 1 (r] R)(n+l)/n- = ---'100CneF(T) .az pCp n (r] R)(n+l)/n - 1
Only the last term on the right-hand-side of (5.25) is a function of r, so we conclude
that the temperature is a function of both rand z, which contradicts our previous
finding that T = T(z). Hence, the model is inconsistent.
We can remove this inconsistency by extending the system to include weak com-
pressibility, as follows
~ ~ ('1r av)r ar ar
~~(r aT) + '1 (av)2r ar ar ar
pep, T)
o
(5.25)
dp
dz
aT
pCpaz
pa(pv)
az
plus boundary conditions at the inlet, outlet and the die wall. The specification of
a (simplified) constitutive relation p = pep, T) is nontrivial and we leave its deter-
mination for future work. Although the model in Stroh et al. (1990) is inconsistent,
their numerical solutions may be valid after all, because their numerical technique
may implicitly add some weak compressibility.
(5.26)
5.6 Conclusions and future work
In this report we have shown that a non-isothermal, no-slip boundary condition model
for extruding polymer flow can explain the observed hysteresis in pressure-flow ex-
periments. Furthermore we have calculated that (especially in the slow flow regime)
while temperature effects may exist, they do not seem crucial to the observed behav-
ior. Instead we believe that the direction taken by the researchers at DOW - attempt-
ing to understand the dynamics at a molecular level near the boundary - is the more
fruitful one.
Acknowledgment
This work started during the 39th European Study Group with Industry at the Uni-
versity of Twente. The problem of polymer extrusion was brought forward by dr. J.
den Doelder of Dow Benelux N.V., Temeuzen, The Netherlands. It is a pleasure to
acknowledge the stimulating discussions with drs. J. den Doelder and J. Molenaar.
Last but not least, we wish to thank the organizers of the study group for organizing
a very inspiring and enjoyable meeting.
66
Thennal Modeling in Polymer Extrusion SWI2000
Bibliography
[1] C.F.J. den Doe1der. Design and implementation of polymer melt fracture mod-
els. Ph.D. thesis. TU Eindhoven, 1999.
[2] J. Molenaar and RJ. Koopmans. Modeling polymer melt-flow instabilities. J.
Rheology. 38(1), pp. 99-109, 1994.
[3] F. Stroh, J. Pita, V. Balakotaiah and D. Luss. Thermoflow multiplicity in a
cooled tube. AICHE Journal 36(3), pp. 397-408, 1990.
67
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