a comprehensive 3d analysis of polymer flow through a conical spiral extrusion die

Fibers and Polymers 2014, Vol.15, No.1, 84-90

84

A Comprehensive 3D Analysis of Polymer Flow through a Conical Spiral

Extrusion Die

Oktay Yilmaz*, Emre K sasöz1, F. Seniha Guner

1, Cagri Nart, and Kadir Kirkkopru

Faculty of Mechanical Engineering, Department of Mechanical Engineering, Istanbul Technical University,

Istanbul 34437, Turkey1Faculty of Chemical and Metallurgical Engineering, Department of Chemical Engineering, Istanbul Technical University,

Istanbul 34469, Turkey

(Received February 6, 2013; Revised May 16, 2013; Accepted May 22, 2013)

Abstract: Several restrictions which are related to extruder machinery and nature of process material exist in the design ofplastic extrusion dies. To this respect, it is very important to consider design criteria and limitations in order to operateextrusion dies at desired production rate and temperature. In the current study, flow field characteristics through a conicalspiral mandrel die are analysed in detail by 3D Computational Fluid Dynamics (CFD) simulations. The effects of operatingconditions such as production rate and temperature on pressure drop through the spiral mandrel die and the occurence of meltfracture are investigated. The temperature dependent viscosity versus shear rate data for grade QB79P (CarmelTech)polypropylene (PP) melt under study are measured by use of rotational and capillary rheometers. Stress terms in themomentum equations are modeled by Generalized Newtonian Fluid (GNF) Model. For this, Bird-Carreau Model is employedas the viscosity model for the polymer melt. 3D CFD analyses provide comprehensive data and understanding with regard toflow behaviour through complex extrusion dies.

Keywords: Conical spiral mandrel die, Polymer extrusion, CFD, Generalized newtonian fluid model

Introduction

Spiral mandrel dies which are used in coextrusion heads

offer some benefits to the manufacturers in the plastics

industry. Spiral mandrel dies provide good thickness uniformity

with a broad range of processing parameters (raw material,

throughput and temperature), short residence times (chemical

property or colour changes), low pressure drop and good

thermal control in film, sheet and pipe production. They

eliminate the weld lines due to spider legs, which support the

mandrel in the annular die of conventional dies. For this

purpose, spiral grooves are typically machined in the mandrel.

The spiral grooves distribute the melt circumferentially to

increase the uniformity of the velocity at the die exit and

help mixing of polymer melt. Number of layers in a coextrusion

process can be increased by adding identical conical spiral

mandrel die geometries such as the one in Figure 1, con-

secutively. Therefore, diameter of a coextrusion head which

consists of conical spiral mandrel dies is smaller than that of

a coextrusion head which consists of cylindrical spiral

mandrel dies. Coat-hanger dies can be employed in conical

spiral dies in order to distribute the polymer melt uniformly

through the entrance section of spiral grooves. The conical

spiral mandrel die shown in Figure 1 can be split into three

sections. The first section is the pre-distribution system

which consists of cylindrical and rectangular channels. The

second one is the coat-hanger type distributor and the last

one is the spiral die section. Design of the distribution

system is crucially important in order to provide uniform

flow through the spiral die inlet. Thus, the spiral die at the

last section can be operated with maximum performance in

terms of balanced velocity distribution at the die exit.

Researches on coat-hanger dies [1-4] and spiral dies [5-11]

have been performed for several decades. In these early

studies, flow inside the spiral and coat-hanger die was

analyzed by one dimensional or two dimensional simplified

approaches. These approaches can be divided into two groups.

The analytical approaches are used for the optimization of

extrusion die geometries in the first group [2-4]. The polymer

melt flow in extrusion dies is generally slow and is of laminar

character. Therefore, one dimensional flow (lubrication

approximation) through the rectangular and circular channels

can be assumed. Mathematical expressions for the extrusion

die geometries such as distribution channel form of a coat-

hanger die are obtained by use of analytical approaches. In

i

*Corresponding author: [email protected]

DOI 10.1007/s12221-014-0084-4

Figure 1. The conical spiral mandrel die under investigation.

Polymer Flow through a Conical Spiral Extrusion Die Fibers and Polymers 2014, Vol.15, No.1 85

the second group of design methods, electrical network

approach is employed for obtaining flow distribution in

extrusion dies [1,6,7,9]. In this method, flow channels of

extrusion die are divided into control volumes and analytical

expressions in [18,20] are used for flow through these

control volumes which have simple shapes such as circular

and rectangular. Flow distribution through extrusion dies can

be determined by solving the resultant system of equations.

The die performance can be optimized by changing the

geometric parameters until satisfactory flow distribution is

achieved. Analytical approaches and electrical network methods

are both relatively easy and fast methods to implement die

design. Hence, these techniques are still very useful for the

preliminary die design. However, a full three-dimensional

simulation of flow in spiral mandrel dies is required for an

accurate analysis and obtaining flow field characteristics in

detail. In contrast to analytical and other simplified approaches,

nonisothermal processes and some rheological characteristics

of polymer melts such as viscoelasticity, elongational viscosity

can be considered in CFD simulations [12-15]. Zatloukal et

al. [12] performed CFD simulations for a flat spiral mandrel

die. When the viscoelastic properties of the polymer melt are

included in numerical simulations, velocity distribution at

the die exit is not changed. Hence, Zatloukal et al. [12]

concluded that incorporating only shear thinning behavior of

the polymer melt used in their study is satisfactory from the

point of view of velocity distribution at the die exit.

Skabrahova et al. [13] studied effects of non-symmetrical

inputs of temperature and mass flow rate prior to the spiral

die on velocity and temperature distribution at the die exit.

The die performance was affected negatively from feeding

of the spiral die entrance non-uniformly. Sun and Gupta [14]

carried out finite element method (FEM) simulations for

polymer melt flow through a cylindrical spiral mandrel die

with star distribution system. In their study, the effects of

elongational viscosity on flow field characteristics were

analyzed. It was concluded that including the elongational

viscosity of polymer melt in CFD simulations resulted in an

increase in pressure drop through the die. On the other hand,

velocity distribution was not affected. Wanli and Xinhou

[15] analyzed the effects of some geometric parameters of a

coat-hanger die on the die performance with respect to exit

velocity distribution and residence time by non-isothermal

CFD simulations. Huang et al. [16] performed FEM simulations

for a coat-hanger die which was designed by using an

analytical method introduced in literature [2]. It is shown

that the analytical design method was successful in terms of

flow balancing at the die exit. Nevertheless, stagnation zones

in the die and the process material effect on the flow

uniformity could only be predicted by CFD simulations.

In the present study, the effects of production rate and

temperature on the design of the conical spiral die in Figure 1

are investigated by FEM simulations with respect to maximum

pressure to be supplied by extruder and limit shear stress for

melt fracture. A comprehensive analysis is carried out on the

flow field characteristics through the die in terms of low

limit of wall shear rate, upper limit of shear stress, flow

uniformity inside the die and pressure drop through the die

which are the crucial restrictions for the design of spiral

mandrel dies.

Materials

A polypropylene random copolymer QB79P from CarmelTech

with a melt flow rate (MFR) of 0.28 g/10 min (230 oC, 2.16 kg)

was used in the present work to investigate the flow field

characteristics in the conical spiral mandrel die. The density

was measured by capillary rheometer according to ASTM

D3835-08 standard for the PP melt and by Archimedes'

principle according to ASTM B962-08 standard for the solid

PP. The densities of the material at room temperature, 210 oC,

230 oC, 260 oC are 910, 790, 783 and 772 kg/m3, respectively.

Shear viscosity versus shear rate data for the polymer melt

were measured by use of an Anton-Paar Physica MCR 301

model rotational rheometer with a 25 mm diameter parallel

plate geometry and a capillary rheometer using capillary dies

of 10, 20 and 25 mm lengths with 1 mm diameter hole for

three temperature values. The Bagley and Rabinowitsch

corrections [17] are applied to the capillary rheometer raw

viscosity data. The Bird-Carreau viscosity model (equation

(10) parameters of the material are given in Table 1.

(1)

Here η and are shear viscosity in Pa.s and shear rate in

sec-1, respectively. η0, λ and n are zero shear viscosity in

Pa.s, relaxation time in sec and dimensionless power-law

index of polymer melt, respectively.

Die Geometry

The geometric parameters of the spiral die section in

Figure 1 and Figure 2 are determined by use of analytical

method suggested by Rauwendaal [7]. The curvature of the

spiral die is neglected in this method. Flow is assumed to be

isothermal and polymer melt viscosity obeys the power law

model. Flow through the spiral die is split into two parts. The

first part flows through spiral groove and the second part

flows through annular channel as flow leaks from spiral

grooves. It is assumed that these two split flows do not affect

each other at the interface. Flow through the spiral groove is

η γ·( )η0

1 λγ·( )n

+[ ]1−n( )/2

-------------------------------------=

γ·

Table 1. The Bird-Carreau model parameters of the polymer melt

210 oC 230 oC 260 oC

ηo (Pa) 42655 33916 18587

λ (s) 2.13 2.71 2.08

n (−) 0.328 0.370 0.390

86 Fibers and Polymers 2014, Vol.15, No.1 Oktay Yilmaz et al.

assumed to be pressure flow in a rectangular channel of

height H and width W with use of a shape factor. Flow through

the annular channel is a pressure flow in slit channel. Pressure

is assumed to be constant at the plane perpendicular to the

spiral die axis. Flow distribution through the die can be

calculated by use of analytical expressions in a step-wise

manner along spiral grooves. Spiral die design is optimized

by solving the flow problem iteratively with the change of

geometric parameters until satisfactory die performance is

achieved.

The die has 12 equally spaced spiral grooves with φ=15o

helix angle at the beginning of the spiral, which gradually

increases along the spiral. The winding angle of the spiral

grooves is 180o. Initial spiral groove height Ho is 12 mm and

spiral groove width W is 8 mm. Initial annular channel

thickness δo is 2 mm and this increases through the die

linearly with an angle of β=1o. Half cone angle α of the

conical spiral mandrel die is 30 o.

The geometric parameters of the coat-hanger type distributors

in Figure 1 are optimized by an inverse design technique

which involves electrical network approach [1] for the flow

paths in the distributor. Four coat hanger dies are used to

distribute the polymer melt to the spiral die. The die land

thickness h is 4 mm and the die land length at the die center

yo is 50 mm. The coat-hanger die has rectangular distribution

channels. The outer diameter D of the conical spiral mandrel

die is 500 mm.

Numerical Simulations

CFD simulations are carried out by use of PolyFlow [21],

a commercially available widely used package as in [22],

which uses finite element method. The mesh generated for

the flow domain of the conical spiral mandrel die is depicted

in Figure 3. A cylindrical annular channel with 50 mm

length is added to the spiral mandrel die exit in order to

obtain fully developed flow at the die exit. The computational

domain is divided into numerous sub-volumes in order to

mesh with hexahedral elements. A few complex sub-volumes

are meshed by tetrahedral elements. Thus, it was achieved a

decrease in mesh number and computational time and an

increase in computational accuracy. The computational domain

has 367112 cells in total with hexahedral dominant elements,

providing good mesh quality. Flow in the conical spiral

mandrel die is incompressible, as it is the case for most of

the polymer extrusion processes where pressure is lower

than 350 bar [19] and flow is of laminar character due to the

very low velocity of polymer melt (Reynolds Number <<1).

Due to the very high viscosity of the polymer melt, inertial

and gravitational forces are negligible and this yields a

balance between viscous and pressure forces. CFD simulations

are performed for isothermal conditions assuming that die

temperature is controlled at a fixed value of 230 oC. Stress

terms in the momentum equations are modeled by Generalized

Newtonian Fluid (GNF) Model [20]. For this, Bird-Carreau

Model in equation (1) is employed as the viscosity model for

the polymer melt. The governing equations of the flow are

given below.

(2)

(3)

(4)

Here, η is the viscosity function in equation (1) and is

the strain-rate tensor components. vi, p and τij are velocity

components in m/sec, hydrostatic pressure in Pa and stress

tensor components in Pa, respectively. The flow has fully-

developed conditions at inlet and exit boundaries in CFD

simulations. No-slip boundary conditions are applied at the

∂vi

∂xi

------- 0=

∂p

∂xi

-------∂τij∂xj

--------=

τij η γ·( )γ· ij=

γ· ij

Figure 2. Section drawing of the conical spiral mandrel die.

Figure 3. The computational domain for the conical spiral mandrel

die.


die walls. Mass flow rates to be supplied to the conical spiral

mandrel die are given in Table 2.

Results and Discussion

Analysis of Flow Field in the Conical Spiral Mandrel Die

A CFD simulation for a specific process condition is

performed in order to evaluate in detail the flow field variables

in the conical spiral mandrel die specifically designed for

use in a coextrusion die in order to produce three-layered

pipes. Each conical spiral mandrel die forms one layer of the

pipe. The pipe diameter and thickness will be 32 mm and

3.6 mm, respectively. Each layer of the pipe will have the

same thickness. In this case, mass flow rate for one layer is

116.9 kg/h for a 20 m/min pipe production rate. Process

temperature is set as 230 oC for a typical process temperature

in PP pipe production.

The streamlines through the spiral mandrel die are shown

in Figure 4. It can be seen from this figure that the polymer

melt is distributed to the spiral die inlet by coat-hanger dies.

Mixing effects of the spiral die can be observed, as the

polymer melt has angular velocity component in the spiral

die section. The velocity vectors in spiral grooved die are

shown in Figure 5. After the coat-hanger die transferred the

polymer melt to the spiral die section, large amount of the

fluid particles is in the spiral grooves initially. Towards the

exit of the spiral die, the spiral groove height decreases and

thickness of the annular channel increases. As a result, some

fluid particles leak from spiral grooves to the annular

channel steadily. All fluid particles flow along the main flow

direction after the spiral die. There is an interaction between

the flow in annular channel and the flow in spiral groove as

seen in Figure 5, this phenomenon can not be taken into

account in analytical methods [5-7] used for solving flow

distribution in spiral dies. The material flow are entirely in

the main flow direction between two spiral grooves in the

annular channel.

The deviation of the local velocity from the mean velocity

with respect to the angular position in the annular channel at

the spiral die exit is shown in Figure 6. The maximum

deviation of the local velocity at the spiral die exit is about

±1 %. The analytical method suggested in [7] and adapted

for the present die calculated maximum deviation about

±2.5 % from the mean velocity at the spiral die exit. The rest

of the die (die land) after the spiral can eliminate the

deviations from the mean velocity at the spiral die exit up to

10 % [8]. Hence, it can be said that the analytical method

used for the design of the conical spiral die is successful for

providing uniform velocity distribution at the die exit.

Figure 7 depicts the contours of velocity magnitude and

Table 2. The mass flow rates through the conical spiral mandrel die

in CFD simulations

Mass flow rate, m.

(kg/h)

29.2

58.5

87.7

116.9

146.1

175.4

204.6

233.8

Figure 4. Stream lines through the spiral mandrel die.

Figure 5. The velocity vectors (a) from coat-hanger distributor

through the spiral die section (b) from spiral die section.


shear rate in spiral die cross section. Both of these variables

have similar variations in the flow domain. The magnitudes

of the variables in spiral grooves are relatively low due to

wider cross section. Between two spiral grooves in the

annular channel, the shear rate and velocity are relatively

high due to narrower cross section. As the diameter of the

conical spiral die and depth of the spiral grooves decrease in

the main flow direction, shear rate and velocity continuously

increase in magnitude. The shear rate in spiral mandrel die

walls is desired to be greater than 5 sec-1. Providing wall

shear rates over the low limit (5 sec-1) is vitally important for

filled polymer melts which have yield stresses to start flowing.

In the case of very small shear rates, the residence time of

the material through the die will be long and polymer melt

may be thermally degraded leading to change in its chemical

structure and colour [19]. The minimum shear rate takes

place in the first spiral groove as seen in Figure 7 and its

value is around 8 sec-1. Therefore, consideration of low limit

of shear rate is necessary in the design process of spiral dies.

Pressure distribution inside the flow domain of the conical

spiral mandrel die is depicted in Figure 8. As the flow cross

sectional area of the die decreases towards the end of the die,

pressure gradient increases. The pressure drop through the

die head should not exceed the extruder maximum pressure

in order to operate the die at the specified production rate

and process temperature. Pressure drop from inlet to the exit

of the coat-hanger die in Figure 1 are calculated 18.52 bar by

electrical network method [1] and 20.74 bar by CFD. Pressure

drop from spiral groove beginning to spiral groove end is

estimated 49.78 bar by analytical method [7] and computed

42.29 bar by CFD. The interface between spiral grooves and

annular channel in spiral die section from which flow leaks

is assumed to be solid wall in the analytical method [7] for

spiral die. Hence, analytically estimated pressure drop through

the spiral die is larger than that computed by CFD.

Malekzadeh et al. [11] reported that Rauwendaal’s method

[7] solution agrees well with FEM results for pressure drop

through a flat spiral die. Analytical approaches and electrical

network methods seem indispensable tools for predicting die

performances expeditiously.

When the polymer melt is exposed to shear stresses above

a certain critical limit at the die walls, especially along the

die land close to the die exit, the plastic material leaves the

die exit having irregular or wavy surfaces. This phenomenon

is called melt fracture in the literature [19]. The melt fracture

limit is mainly dependent on the processed polymer. However,

this limit can be increased by adding plasticizers [19]. In the

Figure 6. Deviation of the local velocity at the spiral die exit with

respect to the average velocity at this cross section.

Figure 7. Contours of (a) resultant velocity and (b) shear rate at the

spiral die cross section which is parallel to the main flow direction.

Figure 8. Pressure distribution through the conical spiral mandrel

die.


present study, melt fracture limit of the PP (grade: QB79P) is

determined by capillary rheometer measurements and its

value is 112000 Pa. The shear stress distribution at the die

walls is shown in Figure 9. The wall shear stresses in the

blue-circled regions (around the spiral die exit and corners in

the flow domain) in Figure 9 exceed the melt fracture limit

of the PP under study. The wall shear stress values of

complex flow channels such as corners cannot be calculated

with analytical approaches several of which can be found in

[18] developed for the design of extrusion dies. Hence, CFD

analyses are to be carried out in order to examine the flow

field characteristics of polymer melts in complex flow channels.

Besides, the wall shear stresses are relatively high in the

annular channel between spiral grooves due to smaller cross

sections in Figure 9. High wall shear stresses at walls can

only be reduced by decreasing the production rate or increasing

the die temperature for an existing die design, but in this

case production cost increases. Thus, the cross sections of

the die should be determined carrefully, in order to operate

the die at the desired temperature and production rate

without having any process limitations.

The Effects of Extruder and Process Material Limita-

tions on Operating Conditions of Extrusion Dies

CFD simulations are performed for different mass flow

rates given in Table 2 and for temperatures of 210, 230 and

260 oC. The throughput-pressure drop curves of the conical

spiral mandrel die are shown in Figure 10 for different

temperatures. Power-law model are fitted for each experimental

data set and the related model parameters are given in Figure

10. The relation between the pressure drop through the conical

spiral mandrel die and mass flow rate is of the following

form.

(5)

Here, n is the power-law index of the polymer melt for

shear viscosity at process temperature as seen in Table 1. R

is the flow resistance of the spiral mandrel die and is

dependent on the die geometry, the process material and

temperature. This relation is of the same form as that for the

simple shaped dies such as circular, slit or annular channels

as suggested in [18]. The pressure drop is proportional nearly

one third power of the flow rate. For process temperature of

210 oC, when the mass flow rate is increased from 29.2 kg/h

to 233.8 kg/h, the pressure drop through the die increases

from 101 bar to 202 bar. This is only two times increase in

the pressure drop against 8 times increase in the flow rate. In

contrast, pressure increase will be 8 times for an 8 times

increase in flow rate in laminar flow of a Newtonian fluid.

This is very beneficial for the point of view design of the

plastic extrusion dies because pressure drop does not change

too much in a broad range of production rates.

The maximum shear stress at the die walls is given for

various mass flow rates and the processing temperatures in

Table 3. The risk of flow instability increases for low

temperatures and high mass flow rates as can be seen from

CFD simulation results in Table 3. As the temperature decreases,

the viscosity of the polymer melt and resulting maximum

P∆ Rm·n

=

Figure 9. Shear stress distribution at the die walls.

Figure 10. The characteristic curves of the conical spiral mandrel

die when it is operated by PP (QB79P).

Table 3. Maximum wall shear stress through the die for different

process temperatures and mass flow rates

Maximum wall shear stress (Pa)

Mass flow rate (kg/h) 210 oC 230 oC 260 oC

29.2 085316 063300 42358

58.5 109060 082827 56325

87.7 125130* 096303 66229

116.9 137618* 107304 74064

146.1 148265* 116546* 80979

175.4 157952* 124537* 87099

204.6 165320* 132046* 92350

233.8 173296* 138604* 97172

*Maximum wall shear stress exceeds the limit shear stress for melt

fracture.


wall shear stress increase. Therefore, the die must be operated

with a high temperature in order to reach high production

rates. In this case, operating cost increases due to increasing

energy consumption for additional heating. Consequently,

designing of flow channels of the spiral mandrel die by taking

into consideration of material property (melt fracture) is of

critical importance for extrusion with desired production rate

at a specific processing temperature.

Conclusion

Low limit of wall shear rate, upper limit of shear stress,

flow uniformity at the die exit and pressure drop through the

die are crucial design criteria for plastic extrusion dies. The

flow field characteristics of a conical spiral mandrel die are

investigated in detail by a CFD simulation in the current

study. In contrast to analytical techniques, CFD analyses for

material flow in complex extrusion dies provide extensive

data and understanding related to design limitations. CFD

analyses can predict the flow shape at intricate sections such

as stagnation and intersecting zones in extrusion dies which

cannot be configured by analytical approaches as pointed

out in [12,16]. CFD data can be used in the design process of

extrusion dies for further modifications. This may prevent a

die operating at off-design conditions which causes increase

in production costs of plastic products. To this respect, during

design process of an extrusion die, preliminary die design is

to be carried out by use of analytical approaches subject to

process conditions. Then, revisions on die geometry must be

made according to detailed analyses of CFD simulation results.

Yet, analytical methods are employed successfully for the

design and optimization of extrusion dies in literature.

Malekzadeh et al. [11] reported that Rauwendaal’s method

[7] and CFD results from literature [12] for a flat spiral die

showed good agreement in terms of pressure drop through

the die and velocity distribution at the die exit. Huang et al.

[16] optimized a coat-hanger die for uniform flow at the die

exit by use of an analytical method introduced in [2] and the

die performance is validated by CFD.

Likewise, pressure drop values estimated by the adapted

analytical approach suggested in [7] for spiral die and by

electrical network method [1] for coat-hanger die agree

satisfactorily with those of CFD results in the current study,

though analytical and electrical network methods do not

yield detailed information about complex flow structures.

The relation between pressure drop and mass flow rate in

the conical spiral mandrel die is of the same form as that for

the simple shaped dies such as circular, slit or annular channels

for the polymer melt rheological properties of which are

modeled by Generalized Newtonian Fluid Model. Unlike in

laminar flow of Newtonian fluids, there is no substantial

increase in pressure drop against considerable increase in

flow rate for the polymer melt flow in the conical spiral die.

This allows operation of extrusion dies in a broad range of

production rates with relatively small changes in extruder

power.

Acknowledgements

We gratefully acknowledge the financial support of

Ministry of Science, Industry and Technology of the

Turkish Republic and Mir R&D Ltd. Co., through grant

number: 00309.STZ.2008-2.

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a comprehensive 3d analysis of polymer flow through a conical spiral extrusion die

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