55 dynamic modeling of blown film extrusion

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Dynamic Mod e l i ng of B lown-F i lm Ex t rus ionJ.CARLPIRKLE,JR., ndRICHARD D. BRAATZ

600 SouthMathews Avenue, Box C-3Universityof lZlinois at Urbana-ChampaignUrbana Rlinois 61801Past dynamic studies of blown-film extrusion have been confined to the stabilityanalysis of the linearized equations. The full set of nonlinear equations comprises asystem of partial differential and algebraic equations with boundary conditions thatvary from author to author. In this paper, the Numerical-Method-of-Lines, whichcombines finite-difference methods with ordinary differential/algebraic equation inte-grators, is used to solve the full system. Appropriate boundary conditions are selectedto give physical results that compare well with experiment. An important boundary

condition is the minimum order reduction condition on the gradient of the bub-ble-tube radius with respect to distance above the extrusion die (the axial position).Transient startups and operational disturbances a re examined. Calculations showthe influence of oscillations in operating conditions such as heat transfer or inflationpressure on the bubble-tube radius and f i lmthickness. Steady-state results obtainedby integrating the transient equations for a sufficiently long time are qualitatively inagreement with experiment, in contrast to pa st simulations of these equations.

INTRODUCTIONlown film extrusion is used to manufacture plas-B ic bags and sheets of thin thermoplastic films

1). Although sign&cant effort has been made in thesteady-state analysis of blown film extrusion, littlehas been attempted other than linearized stabilityanalysis for the dynamic modeling of this process. D y -namic modeling enables the examination of strategiesfor process startup, for handling process upsets, andfor process control.Here the system of PDEs for the dynamic modelingof blown film extrusion is solved using Numerical-Method-of-Lines(NMOL)2).The dynamic and steady-state solutions are presented for the Pearson-Petriemodel (3, ),which has been a mainstay for modelingblown film extrusion of thin films.The NMOL method(spatial discretization of variables and approximationof derivatives by finite differences, followed by time in-tegration to steady-state) avoids the instability ofshooting methods used to solve steady-state equa-tions directly. We show that the NMOL method yieldssteady-state results that are qualitatively differentfrom the simulation results reported for the Pearson-Petrie model by Liu etd 5).Furthermore, our resultsare in good qualitative agreement with the experimen-ta results (6).This finding is important, because thesteady-state simulation resu lts obtained by Liucaused him to advocate an alternative model, thequasi-cylindrical model, to the thin shell model ofPearson and Petrie. This quasi-cylindrical model wasused by subsequent researchers (7).

This paper describes the blown film extrusion pro-cess, presents the dynamic equations for Pearson-Petrie model, discusses some steady-state results (ob-tained from the dynamic model at long times), andexamines the dynamic response of the model to bothstar tup a nd various process disturbances.

DESCRIPTION OF THE BLOWN FILMEXTRUSION PROCESSMolten polymer is extruded through an annular diewhile air is fed through an inner concentric bubble-tube (seeRg. 1 . This internal air causes the cylindri-cal filmto inflate, increasing the radius of the polymerbubble by stretching it, and decreasing the filmthick-ness. Simultaneously, the guide rolls above the dieflatten the film and the nip rolls subject the film totension in the axial (upward from the die) direction.External air supplied from a concentric outer ringcools the film. The resulting temperature reduction in-creases the viscosity of the rising film and eventuallyinduces crystaUization as the temperature drops be-low the melting point of the polymer. The crystalliza-tion, in turn , causes a n additional increase in viscos-ity, and the polymer solidifies.The solidification zone is called the freeze zone orfrost zone (1).Within this region, the rapidly increas-ing viscous stiffness cause s the bubble radius and thefilm thickness to stabilize, changing very little as thefilm heads upward toward the nip roils. The nip rollsand the bubble inflation create an elongating force onthe polymer bubble-tube. Also, the inflating air causes

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Dynamic Modeling of Blown-FilmExtrusion

H +Fig. 1 . Schematic of blown zhextrusion

a circumferential tension on the bubble-tube. The re-sulting biaxial stress can further induce crystalliza-tion, an action termed flow-induced crystallization. Al-though this effect has been included in a recent paper(7).it is neglected in most models of blown filmextru-sion (8,9, 10).Pearson and Petrie and others have developedsteady-state models to describe the blown filmextru-sion process in the limit of very thin films (11, 12). Inthese models, variations of the physical variablesacross the thin film are neglected, leaving the vari-ables as functions of axa position (height above thedie) only. Generally, these models involve the continu-ity equations, momentum equations in the axial andcircumferential directions, an d some type of constitu-tive relation. In later applications, these were coupledwith equations for energy and crystallization kineticsin order to follow the effect of temperature and crys-tallization on t he viscosity (5,13).

PREVIOUS DYNAMIC MODELSThe time-dependent equations for t he thin filmmodel of blawn filmextrusion have been presented byvarious authors (8, 10, 14). The equations were lin-earized in order to pedorm stability analysis, but so-lutions for the original nonlinear equations were notpresented. Owing to differences in numerical methodsand constitutive relations, there is some disagreement

in the results reported by these investigators.

Freeze Zone

Guide Rolls

A PR- 10 cmbecause of the large increase in viscosity associatedwith decreasing temperature and increasing crystal-linity. The blowup ratio ( h a l adius/initial radius) in-creases with inflation pressure AP as expected. For aninflation pressure AP slightly above 270 Pa, the bub-ble becomes unstable and the bubble-tube radius ap-proaches infinity, imulating a burst bubble.Figure 3 gives the f i l mthickness for the conditionscorresponding to Flg. 2. The stretching force causedby increasing inflation pressure AP causes decreasedfilm thickness H . At steady state, film velocity at eachaxial position 2 s given by mass continuity as V =VoHo&/HR, assuming constant density. During tran-sition to steady state, the relationship among V, R,and H is more complicated.

Flgure 4 reports the blowup ratio (r at s = L / Q andthickness reduction ( l/ h ) for the stable steady-statesolutions computed for set values of B and F. For afured inflation pressure B, there is only one steady-state solution that intersects with the takeup ratioline (v = 2.988 at s = L/R,). This implies that the

I I I II I I

Om2.05 10 15 20 25 30

Axial Position 2, cmFYg.2. Bubbletube radius pro ms us. inJlationpressure AP: steady-state,o 463K, L = 2.988.

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J.Carl Pirkle, Jr., and Richard D. Braatz0.100.090.080.070.060.050.040.030.020.01 I I I I I I0.00 1 I I I I I

0 5 10 15 20 25 30Axia l Posi t ion Z, cm

Fig. 3 . Film thicknesspro@s us. nfition pressure AP: steady-state,To 463K = 2.988.

3.02.5

0 2.0

5 .5.-c1dg= 1.0

0.5

0.0

p F=O.148 - Pa

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2Thickness Reduction

Hg. 4. Blowup ratio us. thickness reduction or constant values of injhtwnpressure: steadg-state, o 463 K.

blowup ratio increases monotonically with the inflationpressure. In generating the curves in FUJ. 4, he modi-fied tension F was progressively reduced from the valueindicated on the lower right of each curve to the valueindicated on the upper part of each curve. The modi-fied tension F was a good continuation parameter forgettingaround the turning points in the curves corre-sponding to AP = 250 and 270 Pa.The results shown in Figs. 2 and 4 do not exhibitthe nonintuitive qualitative behavior reported by Liuand coworkers (5, 0, 21)when attempting to simu-late the steady-state Pearson-Petrie model. As the in-flation pressure is increased, our results show an in-crease in blowup ratio rather th an the decreasereported by Liu and coworkers in their simulation

paper (5).Unlike their simulation resu lts, ou r resultsare in agreement with their experimental findings 6).Liu and others (5,22)used a boundary condition atthe die exit (y = 0 at s = 0), a special case of E q 8a(with yo = 0). hich differs from the one we used, Eq8c. This difference in boundary condition may be re-sponsible for their contradictory simulation resu lts.For a given value of the inflation pressure AP,adding a small amoun t of viscoelasticity to the consti-tutive relation has the effect of shifting the blowupratio versus thickness reduction curves upward andto the right. This favors higher blowup ratios andthinner films. From the work of previous investigators(8,22, 3), iscoelasticity is expected to increase sta-bility by the strain-hardening effect.

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Dynamic Modeling of Blown-FilmExtrusionF i g u r es 5 and 6 show the temperature and crystal-linity profiles for the various values of the inflationpressure AP. For inflation pressures of 250 Pa and270 Pa, the flattening of the temperature for interme-diate axial position Z is due to crystallization, whichreleases heat. As indicated in E q 9, the rate of temper-at ur e decrease with respect to axial position Z is

smaller dimensionless film thickness h. As shown inRg. 6, the lower temperatures result in more crystal-lization for higher values of inflation pressure AP. Thisis expected, as E q 14 ndicates that the rate of crystal-lization becomes nonzero when the temperature dropsbelow the melt temperature Om and increases as tem-perature continues to drop.The effect of crystallization on bubble-tube stabiliza-tion is profound. At X = 0.44, film viscosity, repre-sented by Eq 23 and t he parametric values in Table3,is increased by seven orders of magnitude from that ofthe melt emerging from the die.The axial gradient, y = aR/dZ, of the bubble-tuberadius R s plotted in Fig. 7 for various values of infla-tion pressure AP. For inflation pressure AP = 250 and270 Pa, the axial gradient y increases after the filmleaves the die, and then declines rapidly to zero whenreaching the frost zone at 2 > 8 cm. At inflation pres-sure AP = 189.8 Pa, the radius profile remains per-fectly flat for all values of 2 i.e., the axial gradient y =0 uniformly. For inflation pressure AP = 0 and 150Pa, th e axial gradient y is negative as the filmleavesthe die, that is, the bubble-tube contracts. As theaxial position 2 increases, the axial gradient y ap-proaches zero. Some investigators have solved thestea dy-s tate e qua tions for blown film extrusion by

guessing the axial gradient y at the die exit 2 = 0)and using a shooting method to achieve some type ofboundary condition at 2 = L 5, 22). Others havestarted by assuming that the axial gradient y = 0 at 2= L, and using a shooting method to get the correctboundary conditions on the variables at Z = 0 (24,25,26). n our case, and that of Cain and Denn 8).is-cretizing the axia position Z from 2 = 0 to 2 = L andusing finite difference approximations to spatial deriv-atives allows the direct satisfaction of any down-stream boundary condition. As mentioned, the resultsreported here are for the downstream boundary con-dition on the axial gradient y represented by E q 8c.Effect of M a c h i n e Tension

To examine the effect of machine tension on steady-state results, we found that the use of the modifiedtension F = F - Brf2was more revealing than usingF, For a die exit temperature of 463 K an d conditionsgiven in Table 3, the modified tension F was variedfrom 0 to a finite value to determine the range of infla-tion pressures that yielded stable solutions. Corre-sponding to each value of F, there is a lower and upperinflation pressure that allows physical solutions. Abovethe maximum inflation pressure, the radius grows un-controllably, indicating a burst bubble. As shown in.8, the minimum inflation pressure is zero from F= 0 to F = 0.3450, ut rises rapidly to equal the max-i muminflation pressure at F = 0.4377. In the rangebetween F = 0.3450 and F = 0.4377, s AP ap-proaches the minimum inflation pressure from above,the bubble radius collapses to an infinitesimal quan-tity, mimicking a vanishingly thin filament traveling

480%Fm 460

4403 4202i 400

380- 360ii 340

3

0 5 10 15 20 25 30Axial Position Z, m

F ig. 5. Film emperatureprojiles us. hfitionpressure AP: steady-state, To 463 K , = 2.988.POLYMER ENGlNEER lNG AND SCIENCE, FEBRUARY 2OU3,Vof. 43, No. 2 405


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J.Carl Pirkle, Jr., and Richard D. Braakz

0.500.450.400.35

c 0.30r 0.258 0.15(II5 0.20

0.100.050.00

0 5 10 15 20 25 30Ax ia l Position Z, cmFlg. 6. Crystallinity proms us . i n t n ressure AP: steady-state,To 463 K, = 2.988.

at ever increasing speed. For F = 0 to 0.4377,here isno stable solution for inflation pressures either abovethe maximum or below the minimum. Above F =0.4377,here is no stable solution for any inflationpressure.In the negative range of the modified tension, F = 0to -0.13. the maximum inflation pressure continues

to decline, while the minimum inflation pressure risesabove zero. Actually, a solution does exist below theminimum inflation pressure, but it corresponds tonegative values ofF a non-operable condition. For F< -0.13, F is always negative.Use of the modified tension F in Fig. 8 instead of thetension F gave more revealing plots of the allowable

0.200.15**f 0.100.05(3-= 0.004 -0.05

-0.100 5 10 15 20 25 30

Ax ia l Position Z, cmFlg. 7. wialgradientp ro m s us. inJlatbn pressure AP: steady-state,To 463 K = 2.988.

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Dynamic Modeling of Blown-Film ExmLswn

250QLg 200a2 150u)u)PS0cc._z 1- 0050

0

I r I

t +Upper limitI t I I Lower limit1

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5Modified Tension F

PTg. 8. U pper and lower i n t t i o n pressure bounds as afunction of n w d i i ension F.

ranges of inflation pressure AP (or B) as F varied mo-notonically while F sometimes exhibited nonmonoto-nic behavior.Effect of Die Ekit Temperature

To examine the effect of die exit temperature Toupon the process variables, the inflation pressurewasset to AP = 250Pa and the take-up ratio to vL= 2.988.All other conditions being equal, the bubble radius isexpected to grow larger as the die exit temperature Tois increased due to the reduction in viscosity. The ra-dial profiles corresponding to die exit temperature To= 443, 63, nd 468 K are given in Q. 9. The radiusprofiles increase with the die exit temperature To,which, unlike the computed results of Liu (20, 1), sphysically reasonable.Other Eflects:Spatiall9 Variable lnfrcltionPressureand Heat lrtansfer Cmmients

In order to make a one-on-one comparison with thecalculations of Liu et al. (20, 1), he calculations justpresented corresponded to spatially uniform inflationpressure and heat transfer coefficients. For complete-ness, however, we examined some other effects on thecalculated steady-state results.Fit-st, the heat transfer coefficient and the inflationpressure were varied spatially according to the calcu-lated aerodynamic profiles of the external cooling arby Akaike et al. (9). hese investigators used the k-Emethod discussed by Abe et al.(27). heir calcula-tions showed that the exterior surface pressure on thebubble-tube was negative near the die, followed by arise to a positive value and then a decline to nearlyzero. As a consequence, the inflation pressure AP cor-responding to an inside bubble-tube pressure of 270

Pa gauge, had the profile shown in Q. 10. In con-trast, the heat transfer coefficient uh calculated byAkaike et al. increased from a low value of approxi-mately 10 W/(m2K)at the die to a higher value of ap-proximately 190 W/(m2K) in the frost zone (about7- 10 cm) and then declined to a small value down-stream of the frost zone (see Fig. 1 I ) . Sidiripoulos andVlachapoulos (28) alculated similar behavior and at-tributed it to a combination of Coanda and Venturi ef-fects.Thus, the emerging melt near s = 0 would be sub-jected to a lower cooling rate and a higher inflationpressure than occurs in the case of constant uh andAP. This combination of occurrences should threatenthe stability of the bubble-tube a t the entrance zone(near o = 0) as it is subjected to greater stress whilethe viscosity of the melt is at its lowest value. In thecase of the deformation thinning model in Eq 23. thebubble-tube burst at an extrusion temperature at To= 463 K.A lower extrusion temperature, To = 457.46K, was required to keep the bubble-tube stable whileachieving a similar blowup ratio. For a non-deforma-tion-thinning model, e.g., one corresponding to = 0,the bubble-tube is less vulnerable to instability in theentrance zone near h = 0, but the extrusion tempera-ture Tomust be higher, about 488K.Another spatial profile for U,, based on the work ofKanai and White (29). as mentioned by Liu (20, 1).In this expression, uh was held constant at uh,o romZ = 0 to Z = Z,,, and then allowed to drop off asU,,0/Z2.5. Our calculations showed that this drop-offhad very little effect on the results as long as 2 was afew centimeters past the beginning of the frost zone. Atthis point, the filmviscosity was several orders of mag-nitude higher than at Z = 0, and the geometric vari-ables R and H were stabilized. As mentioned earlier.

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J.Carl Pirkle, Jr., and Richard D. Braatz

0.20

1.80

iiI

1.601.40

E 1.20a 1.00I 0.80m 0.60

0.40 I I

0.00 I I I I I 10 5 10 15 20 25 30

Axia l Position Z, mFig.9. Effect of die exit temperature To upon bubble radius profles.

,

0 5 10 15 20 25 30Z, cmFig. 10. Qpical prom of in@aibnpressure resdtingfrom aerodynamics of external cooling air: reference (91.

crystallization played a large role in the increase inviscosity. and a lack of cIystallization would result inless f i l mstability for the Kanaiand White model.Comparisonof Boundary Conditionsfor the xialGradient y

There is some debate over the proper boundary con-dition for the axial gradient y of the bubble radius r.

Here is a demonstration of why the oufflow boundarycondition 8c, which is called th e minimum order re-duction condition by Schiesser (30), as used. Forthe case of inflation pressure AP = 270 Pa and take-up ratio vL = 2.988, Fig. 12 compares steady-stateprofiles of the axial gradient y for boundary conditions8a and 8c. The value of the axia gradient at the die,yo, corresponding to boundary condition 8a s taken

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Dynamic Modeling of Blown-FilmExtrusion

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55 dynamic modeling of blown film extrusion

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