Kutz M. Handbook of materials selection
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1006 PLASTIC PARTS PROCESSING II
data. The identities simply demonstrate that the model is self-consistent; they
cannot address the question of the model’s validity.
For the extensional viscosities, the results (which are not trivial) are
1
–
⫽ 3[
⫹ ( ⌿ ⫹ ⌿ )˙
␥
] (2.44)
321221
⫽ 4
(2.45)
4
1
–
⫽ 6[
⫺ ( ⌿ ⫹ ⌿ )˙
␥
] (2.46)
621221
Although these equations correctly predict the Newtonian low-shear-rate limits
of 3
0
,4
0
, and 6
0
, respectively, they are not explicit in the rate of strain, .˙
More serious is the question of strain-thickening versus strain-thinning behavior.
If 0
⬍⫺⌿
2
/⌿
1
⬍ 0.5, then
3
would be strain thickening, but
6
would be
strain thinning; if 0.5
⬍⫺⌿
2
/⌿
1
⬍ 1, the opposite behavior is predicted. If
⫺⌿
2
/⌿
1
⫽ 0.5, then
3
and
6
would each be strain thinning;
4
is always
predicted to be strain thinning.
Goldsmith Empirical Model. Goldsmith
27
determined high-speed, high-
temperature stress–strain data on two proprietary materials, one of which was
an amorphous polymer; the other was semicrystalline. Using a high-speed video
camera in conjunction with an Instron testing machine, molded ASTM D638
Type M-I tensile bars were pulled at three cross-head speeds and seven temper-
atures. The data were correlated as
2
C [˙ ⫺ (1/ ˙)]
1
⫽ (2.47a)
C
⫹ ˙
2
where
is the engineering tensile stress (based on original cross-sectional area)
and is the extension ratio, (L
⫺ L
0
)/L
0
. He found that for the amorphous˙
material
413
14
C ⫽ (2.07 ⫻ 10 ) exp(⫺0.048T) C ⫽ 8.33 ln (2.47b)
冉冊
12
T
whereas, for the crystalline material,
440
11
C ⫽ (4.5 ⫻ 10 ) exp(⫺0.036T) C ⫽ 14.3 ln (2.47c)
冉冊
12
T
where temperature T is in kelvins. This is an empirical analog to the uniaxial
extensional viscosity,
3
; compare Eq. 2.8.
The author has seen the high-speed films of Denson
3
in which sheets of
polymers exhibited planar extension and uniform biaxial extension. Correlation
of his experimental data would yield equations for
4
and
6
, respectively.
2 INTRODUCTION 1007
Equations of Motion
The equation of motion is
⭸v
T
div{
[grad(v) ⫹ (grad(v)) ]} ⫺
v䡠grad(v) ⫹
g ⫺ grad(P) ⫽
冉冊
⭸t
(2.48)
It may have one-to-three nonzero velocity components—the dependent varia-
bles; hence, there may be one-to-three differential equations to be solved. These
may be functions of one-to-three spatial coordinates and, perhaps, time—the
independent variables. If there is only one independent variable, then the differ-
ential equation is ‘‘ordinary’’ (ODE); if there are more than one, then the dif-
ferential equation is ‘‘partial’’ (PDE).
Velocity Profile and Volumetric Throughput
The equations of motion are first solved for the velocity profiles. These are then
integrated to obtain the volumetric flowrate. If the velocity is dependent on one
direction only, then
Q
⫽兰兰v(x ) dA(x ,x ) (2.49)
123
Product Dimensions
For continuous processes, the product dimensions constitute the cross-sectional
area of the profile. For cyclic processes, the shape and/or size of the mold
halves, and the behavior of the polymer melt, determine the dimensions of the
part produced.
Operating Parameters
The operating parameters are those variables that may be changed in order to
affect the measure of performance. These include the screw speed (rpm), wire
speed (m/s), or roll speed (rpm) for extrusion, wire coating, and calendering,
respectively, and injection pressure (Pa) in injection molding.
2.3 Effects of Temperature
Since plastic parts processing inherently consists of shaping pellets or a preform
of polymer into a different desired shape, either a polymer melt or softened
sheet at a high temperature is converted into a solid object at ambient temper-
ature; hence, these processes are nonisothermal. As discussed below, polymer
melts exhibit viscous heating; thus, every plastics parts process is inherently
affected by temperature.
Viscous Heating
Inherent in shear flow is the phenomenon of viscous heating—the irreversible
generation of thermal energy from the transfer of momentum. The magnitude
of this generation, and the subsequent rate of temperature rise, are given by
1008 PLASTIC PARTS PROCESSING II
dT
: grad(v) ⫽
⌽ ⫽
C (2.50)
P
dt
For a polymer melt with density near 1000 kg/m
3
, heat capacity near 1–2 kJ/
kg
䡠⬚C, and viscosity on the order of 100 Pa䡠s, subjected to shear rates in the
range of 100–1000 s
⫺
1
, the temperature will rise from 0.5 to 100⬚C/s.
Thermal Thinning
The effect of temperature on the shear viscosity is described by the Arrhenius
equation,
E 11
a
(˙
␥
,T) ⫽
(˙
␥
,T ) exp ⫺ (2.51)
再冉 冊冎
0
RT T
0
where E
a
is the activation energy of shear [J/kmol] at and R is the energy˙
␥
constant 8314.4 J/kmol䡠K.
The Energy Equation
The energy equation is
⭸T
div[k grad(T)] ⫺
C v䡠grad(T) ⫹
: grad(v) ⫽
C (2.52)
冉冊
PP
⭸t
The first term is the molecular transfer of thermal energy (heat conduction); its
magnitude is determined by the temperature gradient and the thermal conduc-
tivity of the polymeric material. The second term is the convective transfer of
thermal energy (heat convection); its magnitude depends upon the velocity of
the convecting medium and the gradient of temperature in that direction. The
third is the viscous heating term, Eq. 2.50. The right-hand side is the unsteady-
state term. After solving this equation for the temperature profile, its derivatives
yield the magnitudes of thermal energy fluxes (see Chapter 10 of Bird, Stewart,
and Lightfoot
1
).
2.4 Effects of Pressure
Usually, it is an acceptable approximation to regard liquids as being ‘‘incom-
pressible’’; that is, their density changes little with pressure. However, in injec-
tion molding of parts whose shape and dimensions are critical, such as compact
disks, the effect of pressure, from injection at 10,000–20,000 psi to ejection at
atmospheric pressure, must be taken into account.
P–T–V Relationships for Polymer Melts
Although significant advancements have been made in the modeling of the
P–T–V relationship—an equation of state (EOS)—for polymer melts, the simple
one due to Spencer and Gilmore,
28
(P ⫹ ⌸)(V ⫺ ⍀) ⫽ RT (2.53)
may be sufficient for engineering purposes. Here,
⌸ and ⍀ are parameters that
2 INTRODUCTION 1009
permit one to fit experimental data; ⌸ is sensitive to the pressure at which the
V(T) are measured, whereas
⍀ is not—this is a manifestation of the complex
relationship between P, T, and V.
Although Tadmor and Gogos
29
list several other models, Capt and Kamal
30
have compared the applicability of the Tait and inverse-volume equations for 12
polyethylene resins, using isobaric and isothermal data. The Tait
31
equation,
‘‘originally proposed to explain the compressibility behavior of seawater,’’ is
P
V(P,T) ⫽ V(0,T)1⫺ C ln 1 ⫹ (2.54a)
冋冉冊册
B(T)
2
A ⫹ AT⫹ AT ⫹ 䡠䡠䡠 T ⬍ T
01 2 m
V(0,T) ⫽
(2.54b)
再
V exp(
␣
T) T ⬎ T
0 m
B(T) ⫽ B exp(⫺BT)
(2.54c)
01
(2.54d)
where A
i
, B
i
, and C are data-fitting constants and
␣
is a constant thermal expan-
sion coefficient. The inverse-volume equation, due to Kamal and Levan,
32
is
⭸
c ⫹ dT
2
⫽
⫹ T ⫹ (a ⫹ bT)P ⫹ P (2.55)
冉冊 冉 冊
⬁
⭸T 2
P
⫽
0
where
⬁
is the density at zero pressure and0Kanda, b, c, and d are data-
fitting constants; (
⭸
/⭸T)
P
⫽
0
is also an ‘‘adjustable parameter.’’
Pressure as an Operating Variable
Pressure is sometimes an independently adjustable variable that may be used to
affect the measure of performance; other times it is not. In extrusion, the pressure
developed within the barrel is a dependent variable; it is this pressure that gov-
erns the flow of the melt through the die. In calendering, the pressure is also a
dependent variable.
However, in wire coating, where drag flow is the primary source of momen-
tum, an imposed pressure on the melt is a valuable independent variable that is
used to control the thickness of the coating on the wire. Likewise, in film
blowing, the internal pressure is an independent variable that influences the
diameter and thickness of the resulting ‘‘bubble.’’ Clearly, in injection molding,
blow molding, and thermal forming, the pressure influences the rate at which
parts are made; this may or may not have a significant influence on the cycle
time.
2.5 Extrusion of a Rod as an Example
The extrusion of a cylindrical rod is a continuous process that will serve as an
example to synthesize the fundamental concepts discussed above. (The appro-
priate nomenclature appears in Section 3.)
Accept, without derivation or proof, the equation for the approximate volu-
metric throughput, Q, from the isothermal extrusion of a Newtonian melt:
1010 PLASTIC PARTS PROCESSING II
B⬘P
Q ⫽ A⬘N ⫺ (2.56)
extr
where N is the rotational speed of the screw, P is the developed pressure,
is
the viscosity of the melt, and A
⬘ and B⬘ are constants. Accept, without derivation
or proof, the isothermal Newtonian solution for the volumetric flowrate through
a circular hole die as the result of pressure flow, expressed in the form
CP
Q ⫽ (2.57)
die
where C, a constant, is (
D
4
)/(128L), in which D is the diameter of the hole
and L is the length of that hole (the ‘‘land’’ of the die). When equated, since
Q
extr
⫽ Q
die
for an incompressible fluid, Eqs. 2.55 and 2.56 become
A
⬘C
Q ⫽ N (2.58)
B
⬘ ⫹ C
The density of the extrudate is that of the melt,
m
; hence, the mass flowrate is
m
Q.
It is known
33
that the diameter of such an extruded cylinder, as the melt, will
be 13% larger than the die opening from which it is extruded. After air cooling
(or passing through a water bath), the density of the rod is that of the solid,
s
.
The rod is produced at the mass flowrate of
D⬘
2
s
v
s
/4, where D⬘ is the diameter
of the product and
v
s
is the linear take-up velocity. Hence, for a rod, this velocity
is given by
2
4 A⬘D
m
v ⫽ N (2.59)
s
24
(1.13) 128LB⬘ ⫹
D
s
The take-up velocity necessary to produce a rod of diameter D⬘ is a function
of the geometric factors (A
⬘, B⬘, D, and L), the material properties (
m
and
s
,
but not
), and an operating parameter (N). For this and all other zero-level
solutions (isothermal flow of an incompressible Newtonian melt), there is no
interaction between these three categories. That is, changing the screw speed
has its influence, which depends neither upon the geometric factors nor the
material properties; likewise, changing the temperature, which influences
m
and
maybe
s
, manifests its effect independent of geometric factors or N. This situ-
ation is not true for either nonisothermal flows or non-Newtonian fluids. In the
real world, even the influence of the geometric factors depends upon the material
properties and the operating parameters.
One further comment: Accept, without derivation or proof, the equation for
the power,
ن, expended in the isothermal extrusion of a Newtonian melt:
3 EXTRUSION: SINGLE SCREW 1011
2
A⬘
222
ن ⫽ Ez
N ⫹ A⬘PN ⫽ Ez ⫹
N ⫽ F⬘
N (2.60)
冉冊
B⬘ ⫹ C
where z is the helical length of the extruder screw and E is a constant; hence,
F
⬘ is a constant. Note that a 20% increase in the quantity of rod produced is at
the expense of a 41% increase, (1.2)
2
⫺ (1)
2
, in the power required. Hence, the
rate of return for sales decreases with increased production.
PART I. Continuous Processes: Shear Dominated
In each of the following processes, the isothermal Newtonian solution for the
measure of performance is a function of the relevant geometric factors, material
properties, and operating parameters. The more realistic problems are addressed
where solutions are available. (The nomenclature for each process appears at the
beginning of its section.)
3 EXTRUSION: SINGLE SCREW
Constants and Definitions
A
⫽
mbdw cos(
)[1⫺ (c/b)
2
]F
d
⬇ A⬘
1
–
2
A⬘ ⫽
2
bd
2
cos(
) sin(
)ifm ⫽ 1, c ⬇ 0, and F
d
⬇ 1
1
–
2
b ⫽ Channel depth of the screw in the metering zone
B
⫽
(mb
3
w/z) sin(
){1⫹ (c /b)
3
(w/e)[cos(
)sin(
)]
⫺
2
}F
p
⬇ B⬘
1
––
()
12
B⬘ ⫽
)(b
3
d/z) sin(
)ifm ⫽ 1, c ⬇ 0, and F
p
⬇ 1
1
––
(
12
c ⫽ Clearance between the screw flights and the barrel
C
⫽ WH
3
/12L, rectangular slit of width W and thickness H
⫽
D
4
/128L, cylinder of diameter D
⫽ (
D
4
/128L)ƒ(
), concentric cylindrical annulus with outer diameter D
and inner diameter
⬅
D where ƒ(
) ⫽ [(1 ⫺
4
) ⫺ (1 ⫺
2
)
2
/ln(1/
)]
⬇ (
D
4
/192L)(1 ⫹
⫺
3
⫺
4
), pipe, if
⬎ 0.6
see Tadmor and Gogos
29
for other geometries
d
⫽ Inside diameter of the barrel
D
⬘ ⫽ Diameter of a solidified cylindrical rod; D⬘ ⬇ 1.13D
e
⫽ Width of the screw flights
E
⫽
3
(d
3
/b) sin(
)[1 ⫹ sin
3
(
)]
F
⫽ Ez ⫹ A
2
/(B ⫹ C)
F
⬘ ⫽ Ez ⫹ A⬘
2
/(B⬘ ⫹ C)
F
d
⫽ (16w/
3
b)(1/i
3
)tanh(i
b/2w)
⬁
兺
i
⫽
1,3,5,...
F
p
⫽ 1 ⫺ (192b/
5
w)(1/i
5
)tanh(i
w/2b)
⬁
兺
i
⫽
1,3,5,...
H⬘ ⫽Thickness of a solidified sheet; H⬘ ⬇ 1.20H
l
⫽ z sin(
), axial length of the metering zone; l/d ⬇ 10
L
⫽ Length of the die (in the direction of product flow); the ‘‘land’’ of the die
m
⫽ number of parallel channels in the screw
N
⫽ Screw speed, as ‘‘revolutions’’ per unit time; the ‘‘traversed circumference
per revolution,’’
d, has already been incorporated into the screw constants
P
⫽ Pressure developed by the screw, the pressure drop across the die
Q
⫽ Volumetric flowrate developed by the screw, volumetric flowrate through
the die, each of which is for the melt
1012 PLASTIC PARTS PROCESSING II
S ⫽ Cross-sectional area of the solid extrudate
v
s
⫽ Take-up velocity of the extruded product
V
E
⫽ Cost of electricity in customary units, $/kWh
V
p
⫽ Lineal value of the extruded product, $/ft
V
S
⫽ Mass value of the extruded product, $/lb
w
⫽ (
d/m ) sin(
){1 ⫺ (2c/d) ⫺ [me/(
d sin(
))]}, width of the screw chan-
nel
z
⫽ Helical length of the metering zone channel
⫽ Screw helix angle;
⫽ 17.65⬚ is a ‘‘square pitched’’ screw, where ‘‘lead
⫽ diameter’’
⫽ Newtonian viscosity of the melt
⫽ Density;
m
, . . . of the melt;
s
, . . . of the solid
ن ⫽ Power
Extrusion is the most important continuous polymer processing operation. An
extruder with a die makes a never-ending product with a specific cross-sectional
shape. Extruders are also the upstream equipment for the continuous processes
of wire coating, calendering, fiber spinning, and film blowing. They are also
used to produce the upstream preparts for the cyclic processes of blow molding
and thermoforming.
The extruder proper is a rotating screw within a barrel. The diameter of screw
thread(s)—there may be more than one set of flights—nearly equals the inner
diameter of the barrel; i.e., the clearance is very small. It is the normal stress
functions—Eqs. 2.5 and 2.6—of the polymer melt that keep this cantilevered
screw centered in the barrel to prevent metal-on-metal friction.
The screw has three zones: (1) Near the feed port, the diameter of the root
of the screw is somewhat smaller than that of the barrel, producing a deep
channel to accept solid polymer pellets; solid compaction occurs in this zone in
which air is removed. (2) Following is a zone in which the diameter of the root
of the screw increases considerably; all melting of the pellets must take place
within this zone. (3) Finally, there is a metering zone with a shallow channel;
within this zone the melt is delivered to the die at a constant rate. Within the
metering zone, the pressure increases toward the die.
The turning screw drags—not ‘‘pushes’’—the melt toward the die; the screw
supplies momentum to the melt. But, because the screw flight has a small clear-
ance with the barrel, there is pressure flow of the melt toward the feed throat.
The net flow—drag
⬎ pressure—supplies melt upstream of the die at a relatively
high pressure. Pressure flow then causes the melt to pass through the die, pro-
ducing an object somewhat like the shape of the die opening.
Shirrell
34
derived the equations for both the isothermal and adiabatic extrusion
of a Newtonian melt through an arbitrary die. His results corrected several errors
in the version published by Tadmor and Klein.
35
The approximate forms of the
isothermal equations have already appeared in Section 2.5; the rigorous ones
appear here.
BP
Q ⫽ AN ⫺ (3.1)
extr
3 EXTRUSION: SINGLE SCREW 1013
CP
Q ⫽ (3.2)
die
AC
Q ⫽ N (3.3)
B
⫹ C
2
A
222
ن ⫽ Ez
N ⫹ APN ⫽ Ez ⫹
N ⫽ F
N (3.4)
冉冊
B ⫹ C
As was done for the rod in Section 2.5, for a given profile one must select a
measurable dimension, express its relation to the cross-sectional area, and de-
termine C. Mitsoulis
33
determined that, for Newtonian fluids, the increase in
‘‘thickness’’ of the extruded melt varies from 13% for a solid cylinder (
⫽ 0)
to 20% for a rectangular sheet (
→ 1; hollow concentric cylinders—pipes—
have intermediate values, depending on the nonzero value of
.
For the non-Newtonian extrusion of a rod, one might use Tanner’s equation,
36
1/6
2
D⬘ (⌿ ˙
␥
/
)
121 w
⫽ 0.1 ⫹ 1 ⫹ (3.5)
冉冊
D 8
where the subscript w means ‘‘evaluate this term at the wall conditions.’’ This
equation was used by Horwatt
37
in his FORTRAN program to describe the pro-
cess dynamics and control of the diameter of an extruded rod, with take-up
speed as the manipulated variable.
Abdel-Khalik et al.
38
extended the utility of the Tanner equation by developing
a relationship between
⌿
1
and
, based on the Goddard-Miller rheological equa-
tion of state (see Ref. 2, 1st. ed., Art. 7.5).
The material choice and temperature of extrusion will determine
,
m
, and
s
. The desired size of the extrudate determines v
s
. Analogous to the approximate
solution given in Eq. 2.58, for a rod the take-up velocity is given by
2
4 AD
m
v ⫽ N (3.6)
s,rod
24
(1.13) 128LB ⫹
D
s
Similarly, for a sheet the take-up velocity is given by
2
1 AH
m
v ⫽ N (3.7)
s,sheet
3
1.20 12LB ⫹ WH
s
In general,
m
v ⫽ ƒ(screw A and B; die dimensions) N (3.8)
s,extrudate
s
The selection of an extruder—barrel and screw—to accomplish this task will
determine A, B, and E. The capability of that combination will determine P.
1014 PLASTIC PARTS PROCESSING II
Since an extruder is not a positive-displacement pump, it can be operated with
a blank die (one that has no opening); i.e., Q
extr
⫽ 0. The resulting pressure is
the maximum that may be developed:
A
P ⫽
N (3.9a)
max
B
Operating the extruder without a die (P
⫽ 0) yields the maximum throughput:
Q
⫽ AN (3.9b)
max
Obviously, Q ⬍ Q
max
and P ⬍ P
max
. The conditions at which one should operate
a given extruder-die combination requires the customary optimization of the
value of the extrudate and its production cost. Considering only the ‘‘sales’’
value of the extrudate versus the cost of the electricity to turn the screw,
AC
V
mS
N ⫽ (3.10a)
opt
2
2[A ⫹ (B ⫹ C)Ez]
V
E
or
AC(
/
)(V /S)
ms P
N ⫽ (3.10b)
opt
2
2[A ⫹ (B ⫹ C)Ez]
V
E
In these equations, consistent units, or conversion factors, must be used.
Shirrell’s
34
solution of the adiabatic case was accomplished using a Runge–
Kutta numerical iteration. The three equations for flowrate, power requirement,
and temperature rise were
dP
⫽ (A⬘N ⫺ Q) (3.11)
dz B
⬘
dن dP
2
⫽ E
N ⫹A⬘N (3.12)
dz dz
dT 1 d
ن 1 dP
⫽⫹ (3.13)
冉冊冉冊冉冊冉冊
dz CQ dz C dz
with
⫽
exp[⫺

(T ⫺ T )] (3.14)
00
He
34
also modeled the screw extruder as a polymerization reactor for styrene;
his program was written in Basic. Ray
39
developed an improved FORTRAN
version for the free radical, thermally initiated, polymerization of azoisobutyro-
nitrile.
Chiu
40
has written FORTRAN programs for the pressure flow of Newtonian
and Carreau fluids through three different die shapes—a cylinder, a cylindrical
4 EXTRUSION: TWIN SCREW 1015
annulus, and a rectangular conduit. Three different thermal conditions were
considered: isothermal, constant wall temperature, and adiabatic. The density,
thermal conductivity, overall heat transfer coefficient, low-shear-rate-limiting
viscosity, as well as the Carreau parameter, may be temperature dependent.
These programs solve for the velocity and (for the nonisothermal cases) tem-
perature profiles.
Ishmael
41
has modeled the postextrusion cooling and freezing process as the
extruded melt is turned into a solid rod and chopped into pellets. The successive
heat transfers are from the extrudate to (i) air, (ii) water, (iii) air, (iv) puller-
rollers, and (v) air; this sequence is followed by the pelletizer. Required data
include the polymer density, viscosity, thermal conductivity, heat capacity, and
thermal expansion, each as a function of temperature.
The model predicts the length of the water bath, the final temperature, and
the final inner and outer diameters of the extruded strands. These strands are
hollow because the outer surface is frozen first, and subsequent cooling and
contraction creates a vacuum on the inside. The model is applicable to crystalline
and amorphous polymers; it was confirmed by industrial experiments with iso-
tactic and atactic polypropylene. Although Ishmael
41
used spreadsheet software,
the equations are given and permit one to use an alternative means of calculation.
Cecil
42
showed that repeated processing, as experienced by ‘‘regrind’’—
otherwise scrap material chopped into pellets and sent back to be reextruded—
in the feed, affected the color stability of extruded parts. (Reprocessing also
occurs in injection molding, vide infra.) Cecil measured the yellowness index
(YI) of five grades of polypropylene, without and with various additives, sub-
jected to two or five additional passes through an extruder. The results showed
that: (1) repeated processing increased the YI; (2) small amounts of antioxi-
dant—0.10% for two additional ‘‘cycles’’ and 0.05% for five—reduced the YI,
but larger amounts increased it; (4) other changes in the recipe, such as the
inclusion of a filler deactivator, a secondary stabilizer, and/or pigment, also had
an affect on YI.
4 EXTRUSION: TWIN SCREW
Twin-screw extruders are constructed in a variety of ways: they may be co-
rotating (in the same clockwise or counterclockwise direction) or counterrotating
(opposite directions); they may be nonintermeshing (which means that looking
in the axial direction, the flights of the two screws never overlap), partially
intermeshing, or fully intermeshing. (The book by Janssen, cited in the bibli-
ography, shows examples of these configurations on pp. 6–9; the book by Mar-
telli, also cited in the bibliography, has a more complete classification scheme
on pp. 9–12.)
Tolliver
43
successfully modeled a vented nonintermeshing counterrotating
twin-screw extruder used for the final polymerization of an industrial polymer.
The screws were 6 in. in diameter and 50 ft. long, followed by a short single
screw that produced the extrudate. His FORTRAN program requires knowledge
of (a) the type of element, its flight land width, its volume, and its cross-sectional
area, (b) the screw geometry data (barrel diameter, flight clearance, and helical
angle), and (c) locations of the heating zones. ‘‘Rework and additive ports are
defined in separate [program] statements.’’