Numerical modelling and experimental verification of blown film processing

J. Non-Newtonian Fluid Mech. 116 (2003) 113–138

Numerical modelling and experimental verification of blown film processing S. Muke, H. Connell, I. Sbarski, S.N. Bhattacharya∗ Department of Chemical and Metallurgical Engineering, Rheology and Materials Processing Centre, RMIT University, Melbourne, Vic., Australia Received 17 November 2002; received in revised form 20 August 2003; accepted 20 September 2003

Abstract A non-isothermal viscoelastic Kelvin model was developed to simulate the film blowing process. The predictions made by the Kelvin model for processing characteristics, such as bubble diameter, film thickness and strain rate profiles were compared to predictions from the non-isothermal Newtonian model and actual experimental film blowing data of polypropylene (PP). Reasonable agreement was found between the non-isothermal Newtonian model and experimentally observed bubble characteristics. However, by incorporating the elasticity of the polymer elasticity, using the Kelvin model, predictions were found to significantly improve. The reasonably good agreement between the theoretical predictions from the non-isothermal Kelvin model and actual experimental data may be due to the relatively small Hencky strains and strain rates used in the film blowing conditions in this work. Temperature was found to be the most critical parameter which influenced the film blowing characteristics of PP. A correct estimate of the relaxation time of the polymer is particularly important in giving a reasonably accurate fit with the experimentally observed processing behaviour of the polymer. Experimentally observed bubble stability measurements indicated that the stable operating window for PP increased up to a frost line height of 210 mm beyond which the stability decreased. © 2003 Elsevier B.V. All rights reserved. Keywords: Film blowing; Blown film extrusion; Bubble stability; Rheology; Modelling; Simulation; Film; Polypropylene

1. Background and theory The film blowing process, shown in Fig. 1, is an important industrial process which is used to manufacture thin biaxially orientated polymeric film. The molten polymer is extruded through an annular die at constant mass flowrate and a biaxial extension is produced by internal pressurisation and axial drawing. Air jets through a ring surrounding the bubble provide cooling air. The height above the die at which solidification occurs can be controlled through the air flow rate and negligible deformation of the ∗

Corresponding author. Fax: +61-3-9925-2268. E-mail address: [email protected] (S.N. Bhattacharya). 0377-0257/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2003.09.002

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Nomenclature a A B BUR C Ce Ch Cp d D Fz FLH gi G0 G h H Hc L LFW Mn Mw Mw /Mn MDD MFI p P P Q r t ∇τ T Ta Tc u v X1 X2,3 ZF

bubble radius dimensionless bubble force dimensionless bubble pressure blow up ratio dimensionless circumferential stress dimensionless energy dissipation coefficient dimensionless heat transfer coefficient specific heat of the polymer die diameter diameter of bubble tensile force at the freezeline frost line height relaxation strength corresponding to relaxation time, λi zero shear elastic modulus storage/elastic modulus dimensionless local film thickness local film thickness heat transfer coefficient dimensionless axial stress lay flat width of film number average molecular weight weight average molecular weight polymer polydispersity machine (axial) direction draw ratio melt flow index isotropic pressure distance of edge of bubble relative to a reference line the internal pressure measured relative to the external (atmospheric) pressure volumetric flowrate dimensionless bubble radius dimensionless temperature upper convected derivative of stress temperature ambient air temperature crystallisation temperature of polymer dimensionless velocity of film local velocity of film Arrhenius viscosity function coefficient zero shear temperature dependence function coefficients distance between die exit and freezeline

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Greek letters α dimensionless zero shear modulus β dimensionless zero shear viscosity δ Kronecker delta ε strain ε˙ strain rate η0 zero shear viscosity λ average relaxation time λi relaxation, i, in the relaxation spectrum of the polymer ρ polymer density ρt ,s principal of curvature σ principal stresses on bubble τ deviatoric stress φ energy dissipation function, φ = τij eij ω frequency Subscript 0 refers to conditions at the die exit

Flattened

Nip Rolls

Guide Rolls

∆P Hf

af

Freeze Line

H

XF

ao Air

Ho

Polymer Melt

Annular Die

Air Supply

Fig. 1. Schematic of the film blowing process.

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bubble occurs beyond the freeze line in most processes [1,2]. The final film dimensions are determined by a number of process variables such as the blow up ratio (BUR), which is a ratio of the bubble radius at the freeze line to the radius of the die, and the machine direction draw down ratio (MDD), which is the ratio of velocity at the nip rollers to the velocity of the polymer melt exiting the die. The solidified film is flattened into a double-layered sheet by the nip rollers forming an almost airtight seal at the top of the bubble. After the bubble has been flattened by the nip rollers the flat film is reeled up under constant tension either as tubular film or after slitting into sheet film. Ultimate film properties are controlled by molecular orientation and stress induced crystallisation [3]. A stable bubble in the film blowing process is a requirement for the continuous operation of the process and the production of an acceptable film [4]. In general there are three forms of instabilities or combinations of these reported in literature. These are as follows: 1. Axisymmetric periodic variations of the bubble diameter, known as bubble instability (BI). 2. Helical motions of the bubble, described as helical instability (HI). 3. Variations in the position of the frost line height (FLHI). BI and HI have been reported by a number of investigators [5–9], however, FLHI was only recently reported by Ghaneh-Fard et al. [5,9], after previous authors were using the term “meta stable” [2] which was misleading for time-dependent oscillations in FLH. Results on bubble stability have generally been qualitative until recently when Sweeney et al. [10] utilised a video analysis system as an effective non-contact, real time device for quantifying instabilities during film blowing. Sweeney et al. [10] first proposed the diameter range (Dr ) concept for measuring the degree of helical instability. ¯ and the degree of helical instability (DHI) are then derived from the following The average diameter, D, equations: ¯ = P¯ l − P¯ r , D

(1)

Dmax = Pl,max − Pr,min ,

(2)

Dmin = Pl,min − Pr,max ,

(3)

Dr = Dmax − Dmin ,

(4)

DHI =

Dr × 100, ¯ D

(5)

¯ is the average diameter of the bubble and P¯ the average distance of the bubble from a reference where D line, as shown in Fig. 2. The subscript r denotes the distance of the right bubble edge and l the distance of the left bubble edge from the reference line. The bubble was defined as stable if the DHI was less than 20%, partially helically stable if the DHI was between 20 and 40% and helically unstable if the DHI was greater than 40%. The analysis of bubble stability in film blowing has largely focused on PEs (HDPE, LLDPE and LDPE), mainly due to their superior melt strength in comparison to other polymers such as PP. Only recently have results been reported on the bubble stability of PP [9]. Ghaneh-Fard et al. [9] studied the bubble stability of PP and found PP to have a much smaller stable operating window in comparison to PEs. The focus of this work is to utilise the conventional upward film blowing process to develop a stable set of operating conditions for PP. Then subsequently measure experimental profiles (e.g. bubble radius profile) in the stable operating window to verify a suitable rheological model for film blowing of PP.

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Fig. 2. Typical bubble shape observed for a stable PP bubble (FLH = 210 mm, BUR = 2.6, TUR = 6.7).

1.1. Governing equations of film blowing Pearson and Petrie [11,12] first developed in detail the kinematic and dynamic equations describing fluid flow in film blowing. This was based on the thin shell theory where the thickness of the bubble was small in comparison to the bubble diameter. The kinematic and dynamic analysis of the bubble is discussed below and has provided the theoretical framework for most subsequent studies. For the continuous steady state operation of an incompressible fluid, the law of conservation of mass at any point along the bubble yields the following relationship for volumetric throughput: Q = 2πaHvs = constant,

(6)

where vs is the meridional velocity, Q the total volumetric flow rate through the die, a the local bubble radius and H the local film thickness. Since the problem is axisymmetric, vt (velocity in the transverse direction) is zero and vn (velocity in the normal direction) is not exactly zero since the film is changing thickness, but is negligible, similar to fibre spinning and lubricating flows. The derivative of Eq. (6) with respect to, s, the distance along the film yields a relation between the deformation rates in film blowing: dvs 1 dH 1 da = − vs − vs . (7) ds H ds a ds The left-hand side represents the rate of stretching along the film, while the two terms on the right-hand side are, respectively, the negatives of the stretch rate in the thickness (n) and tangential directions (t). The rate of stretching in each direction is a function of measurable quantities.

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The equilibrium force balance in the normal direction yields P σt σs + , = H ρs ρt

(8a)

where σ s and σ t are the extensional stresses in the meridional, s, and transverse, t, direction, respectively; DP the internal pressure measured relative to the external (atmospheric) pressure; and ρs , ρt the principal radii of curvature in the two directions. It can be shown by simple differential geometry that: [1 + (da/dz)2 ]3/2 , d2 a/dz2
 2 1/2 da ρt = a 1 + . dz

ρs = −

(8b)

(8c)

A force balance in the direction of the axis of symmetry, z, yields Fz = −Pπa2 + 2πaHσs cos θ, where F is the applied tension at z = XF , and
 2 −1/2 da cos θ = 1 + . dz

(9)

(10)

In the analysis of Pearson and Petrie [11,12] inertia, gravity, surface tension and air drag effects are neglected. These are generally realistic assumptions due to the thin film bubble membrane and the viscous forces dominating the process for polymer melts. The governing equations can be easily extended to include these effects by incorporating appropriate physical data. These equations combined with a rheological constitutive equation, relating the stresses to the strains or strain rates in the bubble, result in a series of equations which are solved to yield predictions for various film blowing process characteristics (film temperature, bubble radius and velocity profiles). 1.2. Rheological constitutive equations A summary of the various rheological constitutive equations and their limitations for solving the film blowing problem are shown in Table 1. The simplest viscous model is the Newtonian model which was first applied to the analysis of the film blowing process by Pearson and Petrie [12]. Other viscous models have subsequently been proposed such as the power law model by Han and Park [13] and the crystallisation model of Kanai and White [7]. Liu et al. [14] then made predictions in the film blowing process based on the model developed by Patel and Bogue [15] which accounted for both the non-Newtonian behaviour of polymer melts and crystallisation effects of previous models. These models do not allow for the viscoelastic nature of polymer melts and may explain the poor predictive power in film blowing close to the die exit where extrudate swell exists. The difficulty in modelling the film blowing characteristics of linear semi-crystalline polymers such as HDPE or PP is further complicated by the significant temperature drop in the process and the long neck of the bubble followed by a rapid bubble expansion close to the freeze line [1,16].

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Table 1 Summary of various constitutive equations used in the solution of the film blowing process Author/s

Model description

Limitations

Pearson and Petrie [11,12]

Isothermal Newtonian

Han and Park [13]

Isothermal power law

Kanai and White [6]

Non-isothermal Newtonian with crystallisation Modified non-isothermal Newtonian

Did not incorporate the non-Newtonian flow behaviour of polymer melts Did not account for cooling of bubble and viscoelasticity Did not allow for non-Newtonian behaviour of fluid Did not allow for viscoelastic nature of polymer melt Did not allow for axial curvature of bubble and viscoelastic properties of melt

Sidiropoulos et al. [29,30] Liu et al. [14]

Petrie [18]

Pearson and Gutteridge [17] Luo and Tanner [19] Cain and Denn [32] Wagner [33]

Alaie and Papanastasiou [31]

Cao and Campbell [21,22]

Quasi cylindrical bubble combined with non-isothermal power law with crystallisation effects constitutive equation Non-isothermal Newtonian and isothermal purely elastic model. Effects of gravity and inertia included Non-isothermal elastic model Non-isothermal Maxwell model and Leonov models Marucci model Non-isothermal integral viscoelastic equation with Wagner damping function Non-isothermal integral viscoelastic equation with PSM damping function Non-isothermal Maxwell model extended past freezeline with Hookean elastic model

Did not allow for the viscoelastic response of materials Did not allow for the viscoelastic response of materials Solutions highly unstable, did not account for non-linear viscoelasticity Did not account for multiple relaxation time spectrum Complex, did not accurately estimate stresses at the die exit Complex, difficult to estimate previous shear history of polymer melt particularly at the die exit Highly unstable, does not predict creep flow very well

Pearson and Gutteridge [17] first modelled the film blowing behaviour of PP for the downward blowing process using a purely elastic model similar to that developed earlier by Petrie [18]. A purely elastic model was chosen to best describe the rheological behaviour of the polymer in a semi-molten (solid-like) state. Luo and Tanner [19] more recently used a non-isothermal simple upper convected Maxwell model to describe film blowing of PS and compared the predictions with the experimental results of Gupta [20]. Luo and Tanner’s [19] predictions were in reasonably good agreement with experimental results, particularly for stress and strain rate profiles which are normally not well predicted by the generalised Newtonian models. They state that provided one time constant for the Maxwell model is chosen realistically it will give a reasonably accurate fit to experimental data. This describes the rheological viscoelastic behaviour of polymer melts at small deformations and deformation rates. The Maxwell problem is mathematically far more complex than the Newtonian case. When trying to integrate from the freeze line back to the die exit there was numerical instability of the equations, as described by Petrie [18] and Luo and Tanner [19]. The equations were solved by Luo and Tanner [19] from the die exit up to the freeze line, using a

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fourth order Runge–Kutta numerical technique. This was also found to be highly unstable with converged solutions limited to a small range of operating conditions. The simulations using viscous and viscoelastic models of the film blowing process has traditionally evolved up to the freeze line by assuming that no further deformation occurs beyond the freeze line. The application of viscous and viscoelastic models beyond the freeze line can yield physically incorrect results and sometimes contradictory predictions [21,22]. This led Cao and Campbell [22] to extend the analysis beyond the freeze line using a model known as the viscoelastic–plastic model. Using this model the process is rheologically divided at the plastic–elastic transition (PET). The PET is the position above the die where the deformation changes from a fluid to a solid like elastic deformation. The change in flow mechanism is related to a yield stress criteria. The predicted profiles are physically correct beyond the freeze line, however, there is no significant improvement in the predictions of the bubble deformation zone when comparing with experimental results of Gupta [20] for PS. Since most of the bubble growth and deformation in the bubble occurs below the freeze line, modelling of the film blowing process has focused on improving predictions in this area of the bubble. 1.3. Development and solution of Kelvin model in film blowing In this work, the Kelvin viscoelastic constitutive model was developed, for the first time, to simulate the non-isothermal film blowing process of polymer melts. The steady state bubble in film blowing is typically under applied constant tension above the freeze line due to the nip rollers. This is a classical case of creep deformation behaviour for the bubble. For the Kelvin model, it is a well known fact that creep is predicted well and stress relaxation behaviour is predicted poorly, whereas, in the Maxwell model stress relaxation is predicted well and creep flow is predicted poorly [23,24]. The hypothesis in this work is that film blowing predictions, using the Kelvin model, are improved in comparison to the non-isothermal Newtonian and upper convected Maxwell model solutions. The overall stress in the Kelvin model is the sum of the stresses due to viscous part (dashpot) and elastic part (Hookean spring) of the fluid as described in the following equations: τ = τspring + τdashpot ,

(11)

τij = G0 εij + η0 ε˙ ij ,

(12)

where εij is the deformation tensor and ε˙ ij the deformation rate tensor. The temperature-dependent rheological functions required for the successful simulation of the simple Kelvin model are of the form:    1 1 η0 (T) = η0 (T0 ) exp X1 − , (13) T T0 G0 (T) = X2 + X3 T,

(14)

where η0 (T) is the zero shear viscosity at temperature, T; and G0 (T) the zero shear elastic modulus of the polymer, which is calculated as G0 (T) =

η0 (T) , λ(T)

(15)

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where λ is the average Maxwell relaxation time of the polymer. The values of the constants X1 , X2 and X3 were found to be 4924 K, 1843 Pa and −1.964 Pa K−1 , respectively. The definition for strain acting on an element of fluid in each of the principal directions is given by     v a0 H0 ε11 = ln = ln , (16) v0 aH   H ε22 = ln , (17) H0   a ε33 = ln . (18) a0 The rate of strain is given by the time derivative of strain as   v −dH da − , ε˙ 11 = (1 + a2 )1/2 H dz a dz

(19)

ε˙ 22 =

vH , H(1 + a2 )1/2

(20)

ε˙ 33 =

va . a(1 + a2 )1/2

(21)

Substitution of Eqs. (16)–(18) and (19)–(21) into Eq. (12) yields the following equations for deviatoric stress in each of the principal directions:      H a0 H0 2η0 (T)v a τ11 = 2G0 (T) ln − + , (22) aH (1 + a2 )1/2 H a     2η0 (T)v H H τ22 = 2G0 (T) ln , (23) + 2 1/2 H0 (1 + a ) H     a 2η0 (T)v a τ33 = 2G0 (T) ln + . (24) 2 1/2 a0 (1 + a ) a The total stress for each of the components is related to the deviatoric stresses of the constitutive equations by σij = τij − pδij ,

(25)

where p is the isotropic pressure, δij the Kronecker delta, and τ ij the deviatoric stresses. The fact that the stress at the free surface is equal to atmospheric pressure gives p = τ22 .

(26)

Substitution of Eq. (26) into Eq. (25) yields the following equations for the principal stresses of the bubble: σ11 = τ11 − τ22 ,

(27)

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σ33 = τ33 − τ22 .

(28)

Defining the following dimensionless terms: H , H0 a0 σ33 C= , η0 v0

h=

a x , z= , a0 a0 T − Ta t= . Ta

r=

u=

v , v0

L=

a0 σ11 , v0 η0 (29)

Substitution of Eqs. (22)–(24) into Eqs. (27) and (28) yields the following two dimensionless equations relating the stress in the axial and transverse directions to experimentally measurable quantities:      1 2β(t) 2h r L = α(t) ln 2 4 − + , (30) r h rh(1 + r 2 )1/2 h r     r 2 r 2β(t) h + − . (31) C = α(t) ln h rh(1 + r 2 )1/2 r h The dimensionless governing fundamental film blowing equations were derived from Eqs. (6)-(10), respectively, to yield the following equations irrespective of rheological constitutive equation: L=

(A + Br2 )(1 + r 2 )1/2 , rh

(32)

r =

[hC(1 + r2 )1/2 − 2rB(1 + r 2 )] , A + Br2

(33)

where A and B are dimensional constants for the tensile force and bubble pressure, respectively, defined as  2 F z a0 af A= −B , (34) η0 Q a0 B=

πa03 P . η0 Q

(35)

Eqs. (30) and (31) combined with Eqs. (32) and (33) yield a pair of non-linear ordinary differential equations for h and r as functions of z, the axial distance from the die: 2r  r2 (A + Br2 ) = 2α(t)r2 h(1 + r 2 )1/2 ln(r) + 6β(t)r  + r(1 + r 2 )(A − 3Br2 ),   (A + Br2 )(1 + r 2 ) 1 h r α(t) 2 1/2 =− + rh(1 + r ) ln 2 4 − h 2r 4β(t) r h 4β(t)

(36) (37)

with r = 1,

h = 1 at z =

x = 0, r0

dr =0 dz

at z = Z =

xF , r0

(38)

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also α(t) is the dimensionless zero shear modulus defined as α(t) =

a0 G0 (T) V0 η0 (T0 )

(39)

and β(t) is the dimensionless zero shear viscosity defined as β(t) =

η(T) . η(T0 )

(40)

These equations combined with the dimensionless energy equations of Luo and Tanner [19], are the equations required to solve the simple non-isothermal Kelvin model equations for film blowing. The relationship for dimensionless temperature as a function of measurable quantities, h and r is given by the following equation:      r h r  t = Ce C − L (41) + − Ch rt(1 + r 2 )1/2 , r h r where Ce = η0 Q/2πa0 2H0 ρCp Ta is the dimensionless energy dissipation coefficient and Ch = 2πa0 2Hc / ρCp Q the dimensionless heat transfer coefficient. A comparison between Eqs. (36) and (37) for the Kelvin case and the simple Newtonian case [12] revealed an additional term, α(t), which describes the dimensionless zero shear elastic modulus of the polymer melt as a function of dimensionless temperature. The film blowing equations derived from the simple Kelvin model (Eqs. (36) and (37)), combined with the energy balance, given by Eq. (41), were solved by numerically integrating the equations from the freeze line back to die subject to the boundary conditions given by Eq. (38). A program was written in Maple V5.1 to solve the equations and simulate results for r and h. A fourth order Runge–Kutta scheme was employed to perform the numerical integration. The number of steps in the discretisation process for the numerical solution to the problem was chosen as N = 200. The values of the dimensionless constants A (Eq. (34)) and B (Eq. (35)) are initially guessed to solve Eqs. (36) and (37) and Eq. (41). This process was repeated with improved guesses of A and B until the conditions at the die exit were satisfied (r = 1, h = 1 at x = 0). A Newton iterative scheme was employed to carry out this procedure by setting the maximum number of iterations to six and the step size for the numerical computation of the derivatives to a value of 0.01. This resulted in a final error of less than 0.1% in the iteration. There was quite often instability in the numerical system when trying to integrate the equations which do not give a converged solution. This was the case for the modelled film blowing conditions in this work. It is important that the initial guesses for A and B are relatively close to the actual values for a converged solution to occur. To overcome the numerical instability problems in this work the initial estimates of A and B in the simple Kelvin model were guessed using the values obtained from the Newtonian solution to the problem. Similar numerical instability difficulties at high values of MDD were experienced by Petrie [18] for the elastic model. The simple Kelvin model is in essence mathematically similar to the purely elastic model in that the strain is measured relative to the configuration at the die exit and since there is no reference configuration for a liquid exiting a die the simple Kelvin model is in some sense an approximation as is discussed by Petrie [18].

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2. Experimental 2.1. Materials The material used in this study was a commercial polypropylene (PP) homopolymer-MA3, supplied by UCB Films Pty Ltd. Both the rheological and molecular characteristics of the polymer are shown in Table 2. The polymer studied was a “conventional” Ziegler–Natta catalyst-based reactor material. The molecular weight distributions of the polymer samples were determined using a Waters alliance GPCV 2000 which was equipped with two HT6E columns, RI and viscometer detectors. A Rheometrics SR200 constant stress rheometer, equipped with 25 mm parallel plate geometry, was employed for determining the zero shear viscosity of the polymer. Tests were performed at a melt temperature of 210 ◦ C in a nitrogen atmosphere to avoid any oxidative or thermal degradation. The average relaxation time of the polymer was determined from a spectrum of relaxation times, using Eq. (42). The discrete relaxation time spectrum of the polymer was generated by Rhios V4.0 software using linear viscoelastic G and G data at 210 ◦ C: 2 gi λ (42) λ= i, gi λi where gi (Pa) is the relaxation strength corresponding to relaxation time λi (s). 2.2. Bubble stability measurements A single screw extruder equipped with an annular die (die diameter = 40 mm and die gap at exit = 1.25 mm) was used for extrusion of the polymer melt. All experiments were performed at an extrusion melt temperature of 210 ◦ C and a mass flowrate of 3.4 kg/h. It was found that at these conditions PP could be blown in a stable state. Film blowing bubble stability experiments were performed at three different frost line heights of 120, 210 and 300 mm, whilst the MDD and BUR were varied. The maximum BUR attainable was restricted to 4.2 due to the limited width of the take up rollers and the minimum MDD of 1.95 was needed to avoid sagging of the bubble onto the die lip. A description of the experimental conditions used in the bubble stability analysis is outlined in Table 3. The nip roller speed and the amount of air inside the bubble were then simultaneously adjusted to achieve the BUR and MDD values. Cooling of the bubble was undertaken using a single lip air ring that was located just above the die which directed air at room temperature towards the external surface of the bubble. Experiments were performed by maintaining a Table 2 Rheological and molecular characteristics of polypropylene polymer MA3 Polymer

MA3

ρ (g/cm3 ) MFI (dg/min) η0 (Pa s) λ (s) Mw (kg/mol) Mn (kg/mol) Mw /Mn

0.9 3.5 22800 18.4 454 46 9.9

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Table 3 Experimental parameters for bubble stability measurements Mass flowrate (kg/h) ρ (kg/m3 ) V0 (m/min) Die diameter, d0 (mm) Die gap, H0 (mm) Temperature die 1 (◦ C) Temperature die 2 (◦ C) Temperature extruder z1 , z2 , z3 (◦ C) Die–nip distance (mm) TUR BUR FLH (mm)

3.456 790 0.464 40 1.25 210 210 210 2410 1.95–21.5 0.9–4.2 120, 210, 300

constant frost line height (FLH) whilst the BUR and MDD were varied. The desired FLH value was obtained by varying the cooling air flowrate. The polymer mass flowrate was determined by weighing the amount of film produced at 1 min intervals. At least five of these measurements were conducted to determine the average mass flowrate. The nip roller speed was determined by measuring the number of revolutions made by the rollers per unit time with a stop watch. This was repeated at least three times at several nip roller speed settings so that a relationship between the nip roller setting and nip roller speed could be determined. The amount of machine or axial draw down can then be determined as the ratio of the nip roller speed divided by the die exit velocity, known as the MDD. Alternatively, the MDD can be determined as the ratio of the final film thickness to the die gap value. A Sony digital video camera system was used to record the bubble shape and any oscillations. The recorded images were downloaded onto a PC for analysis using the DV Ezy digital software. The recorded images were then analysed by Autosketch V2 for Windows to obtain the bubble diameter and BUR. The measured bubble diameters in the images were calibrated using a ruler which was placed alongside the bubble as shown in Fig. 2. The BUR was also determined from the lay flat width (LFW) of the film according to 0.636LFW BUR = , (43) d0 where d0 is the die diameter (m). At least 10 samples of the film were obtained at each set of conditions to measure the average lay flat width of the film and average final thickness of the film. The BUR values obtained were in close agreement with bubble radius values obtained from the images. The images were also used to determine any degree of bubble instability. The degree of helical instability (DHI) was determined using the analysis developed by Sweeney et al. [10]. Other types of instability such as bubble instability (BI) and FLH instability (FLHI) were qualitatively described from the images. A more detailed description of the experimental work is given by Muke [25]. 2.3. Stable film blowing process characteristics The online processing study of the polymer melt between the die exit and freeze line were carried out for a stable set of processing conditions as outlined in Table 4. The velocity profile in the axial

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Table 4 Experimental parameters for stable bubble profile measurements Mass flowrate (kg/h) Melt temperature, T0 (◦ C) Die diameter, d0 (mm) Die gap, H0 (mm) BUR TUR FLH (mm)

3.9 210 40 0.75 3.1 4.3 340

direction was measured by a standard tracer technique, similar to that employed by previous authors [7,26]. The axial velocity profile data, shown in Fig. 10, was recorded from five experimental runs. The error range for measuring the velocity profile, using the particle tracer technique, was approximately 8%. The velocity data were smoothed by splitting the data into two regions, lower and upper. The lower region, <240 mm from the die exit, where a slow increase in velocity was observed, was fitted using a third order polynomial. For the upper region, >240 mm from the die exit, where the velocity increased rapidly and then levelled off, a fifth order polynomial was used. Carreau et al. [27] also smoothed film blowing velocity data by using fifth order polynomial in the upper region. However, a second order function was found to give a better fit in the lower region. The velocity above the freezeline was measured from the nip roller speed (mm/s) and was found to be in good agreement with the value above the freeze line from Fig. 10. The experimental bubble diameter profile, shown in Fig. 7, was also simply determined using the video images of the bubble. The measured bubble diameter profile of the bubble was measured at steady state conditions for five different experimental runs. The freezeline was estimated to be approximately 340 mm using the video images and was experimentally verified from Fig. 7, where there was a constant bubble diameter with increasing distance from the die exit. Data below 50 mm from the die exit were not obtainable due to the air ring situated just above the die exit. The bubble diameter data were smoothed by fitting a sixth order polynomial function over the whole data range. Errors in the bubble diameter measurement were generally found to be less than 5%. It is important to note that the polynomials can only be used for smoothing the raw experimental data over the range of data applicable in the experiment. For example, extrapolation of the curves in Fig. 7 back to the die exit (axial distance = 0) does not yield the correct die diameter. The film thickness profile as a function of the axial co-ordinate was calculated using an indirect method based on the law of mass conservation, according to Eq. (6). By simply knowing the velocity and radius of the bubble at any point along the bubble the thickness profile can be calculated. This technique assumes that the temperature dependence of the density is neglected as a function of axial distance. The error resulting from the constant density assumption in thickness calculations should not be more than 4% [5]. The calculated final film thickness value above the freeze line was found to be in relatively good agreement with averaged experimentally measured values of the final film thickness using a micrometer. The temperature measurements were carried out using an infrared pyrometer (Model IR-TA manufactured by Chino Corp.). The instrument absorbs the infrared radiation in a wavelength of 3.43 ␮m. Instrument calibration was performed by adjusting the emissivity of the pyrometer so that the measured temperature was in agreement with that of a PP melt bath which was controlled at 190 ◦ C in an air circulated oven using a thermocouple. The emissivity was held constant at 0.95 during the bubble temperature

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measurements. The difficulty with such measurements is that emissivity is a function of thickness and is complicated by the crystallinity of the polymer occurring near the frost line. As yet there is no accurate device for measuring the temperature of the bubble during stretching. The axial (machine) and circumferential (transverse) direction strain rate profiles of the bubble were calculated from Eqs. (44) and (45), by taking derivatives of the smoothed velocity and bubble diameter data, respectively. These are good approximations of Eqs. (19) and (21) since the rate of change of bubble radius in the machine direction is small: ε˙ 11 =

dvz , dz

(44)

ε˙ 33 =

vz da . a dz

(45)

3. Results and discussion 3.1. Bubble stability The bubble stability behaviour of polymer MA3 was carried out at three different frost line height positions—120, 210 and 300 mm. In general three forms of instabilities and combinations were observed: (1) axisymmetric periodic variations in the bubble diameter, known as bubble instability (BI); (2) helical motions of the bubble, known as helical instability (HI); and (3) variations in the position of the solidification line, known as frost line height instability (FLI). Recently Ghanneh-Fard et al. [9], observed similar instabilities for PP by performing experiments at constant MDD. At low BURs for all frost line heights i.e. by inflating the bubble with a small amount of air, bubble instability was observed. The magnitude of the diameter fluctuations increased with time and eventually led to bubble breakage as reported by Minoshima and White [28]. A frost line height instability was observed for different operating conditions for PP. The FLH fluctuated between an upper and lower limit (±50 mm) and the diameter of the bubble varied slightly following the FLH fluctuations. Furthermore, this was accompanied by a significant variation of film thickness, which was also found by Fleissner [4]. Minoshima and White [28] observed FLH instability by observing fluctuations in the bubble tension. FLH instability usually grew with time and then combined with helical instability and eventually caused the collapse of the bubble. Helical instability was observed at higher BURs for PP. A helical motion developed between the die and the nip rollers. This normally developed when the FLH moved from the upper limit to the lower one. This was the predominant form of instability for PP. Fig. 3 shows the stable bubble operating window for MA3 at a relatively low frost line height of 120 mm. It was found that a low value of TUR is required to maintain a stable bubble. The bubble is stable in this space for 2.2 < BUR < 3. This was in agreement with Ghaneh-Fard et al. [9] who found that at low constant MDD ratio the bubble was stable at low frost line heights for a range of BURs. This particular FLH showed the smallest window of stable conditions. As frost line height increased the stable bubble at low MDD disappeared and HI became evident. At an FLH of 210 mm the stable operating conditions were restricted between 1.2 < BUR < 3.5 and 5 < MDD < 8 (c.f. Fig. 4). At higher MDDs helical instability was observed, sometimes in conjunction with other forms of instability resulting in significant variations of film thickness. At higher BUR ratios

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Stable BI+FLH 20

HI BI+HI Pr

TUR

15

10

HI

BI

5

0 1.0

1.5

2.0

2.5

3.0

3.5

BUR

Fig. 3. Bubble stability behaviour of PP MA3 at frost line height = 120 mm.

25

Stable HI+FLH BI HI BI+HI

TUR

20

15

10 BI

5

0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

BUR

Fig. 4. Bubble stability behaviour of PP MA3 at frost line height = 210 mm.

4.5

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129

Stable HI+FLH BI HI PHI

20

TUR

15

10

BI

5

0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

BUR

Fig. 5. Bubble stability behaviour of PP MA3 at frost line height = 300 mm.

HI was generally observed in conjunction with FLH instability. At an FLH of 300 mm the stable operating conditions were restricted between 1.2 < BUR < 2.8 and 5 < MDD < 10 (c.f. Fig. 5). As frost line height increased the maximum stable MDD increased and the maximum stable BUR decreased. Overall, it was observed that PP had the widest stable operating window at a FLH of 210 mm. The results here suggest that the stable operating window increased as the FLH increased up to an upper limit of 210 mm. Above this FLH the stable operating window decreased. This was in agreement with previous work by Ghaneh-Fard et al. [9] who found that increasing the FLH increased the range of stable operating conditions. This was in disagreement with Minoshima and White [28] who suggested that increasing the FLH value decreased the range of stable operating conditions. 3.2. Simulations of the film blowing process In this work the blown film process characteristics were simulated using two existing models (Newtonian and Simple Maxwell model) and for the first time, a model incorporating the viscoelastic Kelvin constitutive equation. The simulated results were compared with experimentally measured profiles for the film blowing conditions shown in Table 4. The non-isothermal Newtonian equations were solved by simply setting the elastic component of Eqs. (36) and (37) to zero. The non-isothermal Newtonian model equations are equivalent to Petrie’s earlier model [18]. The non-isothermal Newtonian model is being used for the first time model to predict the film blowing characteristics of PP. The solution of the Newtonian model equations was performed using a similar numerical solution technique to the Kelvin model described earlier.

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Fig. 6. Curves of storage modulus for polymer MA3 at T = 210 ◦ C.

An additional dimensionless alpha parameter (Eq. (39)) was required with the Kelvin model to describe the influence of temperature on the zero shear elastic modulus of the polymer. The elastic modulus of the polymer was found to be a weak function of temperature and was described by a linear relationship (c.f. Eq. (14)). The values of the parameters used to describe the temperature dependence of elastic modulus were α1 = 9.94 × 10−5 , X2 = 1843 K, X3 = −1.9638 Pa K−1 . Provided the temperature dependence of the rheological properties is described accurately the model should predict results quite well. Fig. 6 shows the computed value of G calculated from the following expression: G (ω) =

η0 λω2 1 + λ2 ω 2

(46)

for an average relaxation time of λ ≈ 8.48 s, in comparison to the experimentally measured storage modulus at T = 210 ◦ C. The average relaxation time calculated from the relaxation spectrum of the polymer, using Eq. (42), gave a reasonable fit of the storage modulus over the relevant range of rates (0.1–1 s−1 ) applicable in this work. Luo and Tanner [19] have shown that provided the time constant is chosen realistically the predictions should be quite reasonable. The Maxwell model equations of Luo and Tanner [19] were developed to simulate the film blowing conditions in this work. However, satisfactory convergence of the Maxwell model for the film blowing conditions in this work could not be obtained. The difficulties in obtaining stable solutions to the Maxwell model at higher values of MDD have been noted by Luo and Tanner [19] and Petrie [18]. However, a more robust model such as the viscoelastic Kelvin model was successfully solved for the film blowing conditions in this work. The various simulation parameters for the Kelvin model are outlined in Table 5.

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Table 5 Parameters used for the Newtonian and viscoelastic Kelvin simulations

Ta (◦ C) Ce Ch α1 X1 X2 X3 t0 tc X ri hi ti rf hf A B tf

Isothermal Kelvin model

Non-isothermal Kelvin model

Non-isothermal Newtonian

298 0.00002234 0.00958744 0.0000993635 4924 1843 −1.9638 0.62131 0.32885 17 1 1 t0 3.12 0.050869887 0.52249276 0.2035708592 10

298 0.00002234 0.021214 0.0000993635 4924 1843 −1.9638 0.62131 0.32885 17 1 1 t0 3.12 0.050869887 0.83885 0.38773 0.32885

298 0.00002234 0.0203981 0 4924 1843 −1.9638 0.62131 0.32885 17 1 1 t0 3.12 0.050869887 0.46278 0.29532 0.32885

A possible explanation as to why the Kelvin model can be successfully solved for various MDD, BUR and FLH’s is that the boundary conditions for the Kelvin model are in strain whereas the Maxwell model requires boundary conditions in stress, which is normally very difficult to estimate at the die exit because of the previous shear history of the melt. For the film blowing conditions in this work it cannot be ascertained whether the Maxwell model can yield better predictions than the Kelvin model. 3.3. Comparison between model predictions from bubble simulation and actual experimentally observed bubble Verification of predictions was performed by comparing simulated results with experimental film blowing process characteristic profiles for PP. The viscoelastic Kelvin model was utilised to overcome shortfalls of the purely viscous Newtonian model for describing the blown film process. Fig. 7 shows the predicted diameter profiles of the bubble for the non-isothermal Newtonian model and the Kelvin (isothermal and non-isothermal) models. The Kelvin model yielded improved predictions in comparison to the predictions given by the non-isothermal Newtonian model. Furthermore, the non-isothermal Kelvin model predictions agreed much more closely to the experimentally observed bubble shape. Sidiropoulos et al. [29,30] found that improved bubble diameter predictions were possible by modifying the Arrhenius viscosity function to allow for an infinite viscosity at the freezeline. However, non-isothermal Newtonian model predictions had failed to accurately describe other process characteristics such as the thickness profile which did not allow for the extrudate swell of the polymer melt exiting the die. The experimentally observed bubble shape, shown in Fig. 7, showed little change in bubble diameter up to a distance of 150–200 mm from the die exit. This was essentially the bubble neck-forming zone. Beyond 150–200 mm from the die exit the rate of bubble expansion suddenly increased and then decreased again

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Fig. 7. A comparison between the actual bubble diameter profile and predicted bubble profiles from different models.

toward the frost line height forming a point of inflection in the bubble shape at approximately 220 mm from the die exit. The frost line height was verified from visual inspection and was cross checked as the point at which the bubble diameter remained constant. A comparison between the Kelvin and Newtonian models revealed the effect of incorporating elasticity of the polymer in the modelling of the film blowing process. The prediction of bubble diameter from the Kelvin model suppresses the bubble expansion. This is reasonable from the argument that G0 (the zero shear modulus) acts as a solid like or elastic parameter in the Kelvin model and that stretching resistance increases with stretch rate. Obviously, the larger G0 is, the more solid-like will be the response of the material. Therefore, the effect of fluid elasticity is to reduce the bubble expansion. As real polymer film blowing operations are characterised by rapid changes of temperature, it is clear that realistic quantitative predictions of the process can only be obtained if the effect of temperature is included. A comparison between the isothermal and non-isothermal bubble diameter profiles showed that the temperature is the most critical parameter for influencing the bubble shape, even more so than the rheological constitutive equation of state. This was also observed by other investigators of the blown film process [1,31,32]. Fig. 8 shows a comparison of the predicted temperature profiles and the measured temperature profile in the film blowing process. The measured temperature profile was found to be inaccurate by simply measuring with an infrared temperature measuring device. The measured temperature of the polymeric film was found to be significantly lower than the predicted temperature profile and was significantly lower than the crystallisation temperature of the polymer at the freeze line. As the thickness of the film decreased (i.e. increased distance from the die exit), the lower measured temperature deviated further from the predicted value. This may be due to the infrared device measuring the influence of the lower background air temperature of the thin polymeric film. The accurate measurement of temperature

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133

220

Temperature (˚C)

200

Measurement Isothermal

180

Non isothermal 160

140

120

100 0

50

100

150

200

250

300

350

400

Distance from die exit (mm)

Fig. 8. A comparison of the predicted and measured temperature profiles of the film blowing process.

is further complicated by the fact that the emissivity is difficult to measure as the polymer crystallises and there is a significant reduction in film thickness. The main focus of this work was to improve film blowing process predictions by verifying that the Kelvin model predicts creep flow much better and by incorporating the effect of elasticity of the polymer melt improved predictions of process characteristics were achievable. Fig. 9 shows the film thickness profile predictions from the models in comparison to the measured film thickness profile. The measured film thickness decreased exponentially with increasing distance from the die exit. The non-isothermal Kelvin model accurately predicted the film thickness variation between the die exit and final film thickness at the freeze line in comparison to the non-isothermal Newtonian case. The improved prediction of film thickness profile for the non-isothermal Kelvin model is particularly important for polymer processing. Usually the film blowing process parameters are adjusted to yield a specific final film thickness and end use properties. The Kelvin model can be used to estimate the final gauge thickness of the polymeric film for different film blowing parameters by simply measuring its rheological parameters (G0 (T) and η0 (T)). The higher film thickness at the die exit was determined by allowing for the extrudate swell of the polymer. In the case of a Newtonian fluid the thickness at the die exit should correspond to the die exit gap. However, the extrudate swell thickness at the die exit is not determined by the models and for a true comparison between the two models the same initial thickness was used. Interestingly, the non-isothermal Kelvin models showed an increased degree of thinning in comparison to the isothermal predictions. This behaviour is “counter intuitive” in a sense that the effect of bubble cooling in the non-isothermal cases should increase the resistance to draw down the polymer film to thinner gauges. The isothermal isothermal Kelvin model showed an inflection point in the thickness profile at approximately 260 mm from the die exit, which corresponded to the rapid increase in the bubble diameter profiles at a similar distance from the die exit.

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Fig. 9. A comparison between experimentally observed film thickness profile with predicted values from different rheological models.

The experimentally observed velocity profile of the bubble between the die exit and freeze line was compared to predictions from the Newtonian and Kelvin models in Fig. 10. The non-isothermal Kelvin model predicted the actually observed velocity profile reasonably well. In the neck-forming zone both models predict an approximately linear increase of velocity with distance from the die exit. The non-isothermal Kelvin model reasonably estimated a rapid increase in the velocity (acceleration) of the polymeric film in the sudden expansion region of the bubble. A more crucial test of the models can be investigated by comparing the stresses and stretch rates formed in the bubble. The stresses were not measured in this work, however, a comparison between experimentally determined strain rates, calculated from Eqs. (44) and (45), were made with theoretical predictions from the Newtonian and Kelvin models, as shown in Fig. 11. Measured axial and circumferential bubble strain rate data are seldom shown in literature. Recently, Ghaneh-Fard et al. [5,9] measured the strain rate profiles of several different types of polyethylenes at different film blowing conditions. They found that the peak axial (machine) direction strain rates were generally higher than the peak circumferential (transverse) strain rates. This was also found in this work and is believed to be related to the higher stretch ratio in the axial direction in comparison to the transverse direction. An interesting finding in this work was that the peak machine direction strain rate occurred further from the die exit than the position of the peak transverse strain rate. Ghaneh-Fard et al. [5,9] found that the peak machine direction strain rate was shifted to further distances from the die exit as the BUR was increased. At a BUR of 2.5 they found that the peak value of strain rate occurred at the same distance from the die exit. The higher BUR of 3.12 and lower MDD of 4.3 in this work may explain the later peak in the transverse direction strain rate.

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Fig. 10. A comparison between experimentally observed velocity profile with predicted values from different rheological models.

Fig. 11. A comparison between theoretical and experimental strain rate profiles.

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Both the Newtonian and the Kelvin models fail to accurately predict the strain rate profiles of the PP bubble. The Kelvin model does however qualitatively describe the strain rate profiles of the bubble reasonably well in comparison to the Newtonian case. The Newtonian model qualitatively predicts the machine direction strain rate in the neck-forming region of the bubble and correctly predicts the position of the peak in the rapid expansion region of the bubble towards the freeze line. However, the Newtonian case fails to qualitatively predict the correct position of the peak in the transverse strain rate and over predicts the value of the transverse strain rate profile throughout. The Kelvin model on the other hand predicted a similar peak value in the transverse direction strain rate but slightly further away from the die exit. In the machine (axial) direction the Kelvin model predicted a rapid increase in the strain rate that qualitatively agreed with the observed situation in the region where the bubble begins to rapidly deform. The strain rate simulations shown in Fig. 11 show similar behaviour to the earlier modelling work of Petrie [18] for the non-isothermal Newtonian model. The results of Petrie [18] showed a non-zero machine direction strain rate at the freeze line which was also the case in this work. The work of Sidiropoulos et al. [29,30] showed that if a modified viscosity function is used which yields an infinite viscosity at the freeze line, a zero strain rate can be obtained at the freezeline. This is not predicted correctly by using the Arrhenius temperature dependence viscosity function. As mentioned earlier the film blowing equations for the Newtonian and Kelvin models were numerically integrated from the freeze line back to the die exit by solving for the dimensionless bubble pressure B and dimensionless parameter A to meet the required boundary conditions at the die exit and freezeline. Table 6 shows the calculated dimensionless pressure parameter (B) and the dimensionless parameter (A) determined for the different modelling cases. Table 6 also shows the predicted bubble pressure and tensile take up force of the bubbles. A comparison of the Newtonian and Kelvin models showed that the bubble pressure increased by incorporating the elasticity of the polymer in the Kelvin model computations. This is understandable from the point of view that the zero shear modulus acts as a solid like parameter which suppressed the bubble expansion rate and required a greater bubble pressure to attain the same blow up ratio. The non-isothermal cooling of the bubble seemed to have the most significant effect on the bubble pressure. A higher blow up pressure was obtained in the non-isothermal case which is understandable from the perspective that there is a greater resistance to bubble deformation as the bubble cools down in the non-isothermal case. Similar behaviour was also shown for the axial take up force. A larger take up force was predicted by the Kelvin model in comparison to the Newtonian case. The increased resistance to draw down the polymer was due to the solid like elastic parameter, G0 , incorporated in the Kelvin model. A higher take up force was also observed for the non-isothermal cases where a higher resistance to flow is expected. The Kelvin model developed in this work showed good agreement with many of the physically observable film blowing characteristics for PP such as the sudden bubble expansion. However, one must keep in perspective that the simple Kelvin model models fail to theoretically describe the non-linear viscoelastic Table 6 Data obtained from numerical modelling computations

Non-isothermal Newtonian Isothermal Kelvin Non-isothermal Kelvin

A

Fz (N)

B

P (Pa)

0.46 0.52 0.84

3.16 2.37 4.38

0.30 0.20 0.39

221 153 292

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rheological behaviour of the polymer melts in the film blowing process. The reasonably good agreement between the theoretical predictions from the non-isothermal Kelvin model and actual experimental data may be due to the relatively small Hencky strains and strain rates used in this work. Almost equal biaxial extensional flow was present in this study with relatively small values of machine direction stretch ratio (axial stretch ratio) which was approximately 4.3 and the blow up ratio (transverse stretch ratio) which was 3.12. The extension of the simple Kelvin model to the generalised Kelvin model for multiple relaxation times would incorporate full linear viscoelasticity of the polymer melt and should yield improved predictions. Future work should be carried out incorporating the multiple Kelvin model and non-linear viscoelastic effects in film blowing predictions. 4. Conclusions A simple viscoelastic Kelvin model was used for the first time to simulate the conventional film blowing process. The kinematic and dynamic expressions of film blowing combined with the Kelvin rheological constitutive equation were solved using a Maple software program which performed numerical integration based on a finite element approach. It was found that the non-isothermal Kelvin model predictions agreed much more closely to the experimentally observed film blowing characteristics of PP in comparison to the non-isothermal Newtonian model. In particular the film thickness profile predicted from the non-isothermal Kelvin model was in good agreement with the experimentally observed thickness profile. The results of this work also indicated that the stable operating conditions of the bubble increased with frost line height (FLH) up to an upper limit of 210 mm. Above this FLH the stable operating window decreased. Acknowledgements The authors wish to acknowledge the assistance of UCB Films Pty Ltd for supplying the polymer and for providing a postgraduate scholarship to one of the authors (SM). The authors would also like to thank Associate Professor J. Shephard at the Department of Mathematics, RMIT University, for helpful suggestions and comments in the modelling work. References [1] J.F. Agassant, P. Avenas, J.Ph. Sergent, P.J. Carreau, Polymer Processing Principles and Modeling, Carl Hanser, Munich, 1991. [2] C.D. Han, Rheology in Polymer Processing, Academic Press, New York, 1976. [3] J.M. Dealy, K.F. Wissbrun, Melt Rheology and its Role in Plastics Processing, Van Nostrand Reinhold, New York, 1990. [4] M. Fleissner, Elongational flow of HDPE samples and bubble instability in film blowing, Int. Polym. Process. 2 (1988) 229. [5] A. Ghaneh-Fard, P.J. Carreau, P.G. Lafleur, Study of kinematics and dynamics of film blowing, Polym. Eng. Sci. 37 (1997) 1148. [6] T. Kanai, J.L. White, Dynamics, heat transfer and structure development in tubular film extrusion of polymer melts: a mathematical model and predictions, J. Polym. Eng. 5 (1985) 135. [7] T. Kanai, J.L. White, Kinematics, dynamics and stability of tubular film extrusion for various polyethylenes, Polym. Eng. Sci. 24 (1984) 1185. [8] C.D. Han, J.Y. Park, Studies on blown film extrusion. 3. Bubble Instability, J. Appl. Polym. Sci. 19 (1975) 3291.

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