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A slit die design for casting plastics sheets

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This research objective is to investigate a simple way to design a plastics coat-hanger die that conveys the uniform outlet velocity and pressure of a power-law fluid across the die width. A coat hanger die with a circular manifold and parallel die
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    AIJSTPME (2010) 3(1): 35-45© King Mongkut’s University of Technology North Bangkok Press, Bangkok, Thailand    35   A Slit Die Design for Casting Plastics Sheets  Arunworradirok S. Polymer Research Center and Department of Mechanical and Aerospace Engineering, Faculty of  Engineering, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand   Kolitawong C.* Polymer Research Center and Department of Mechanical and Aerospace Engineering, Faculty of  Engineering, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand  Email address: ckw@kmutnb.ac.th    Abstract This research objective is to investigate a simple way to design a plastics coat-hanger die that conveys theuniform outlet velocity and pressure of a power-law fluid across the die width. A coat hanger die with acircular manifold and parallel die land for a power-law fluid is investigated. Two charts of dimensionlessgroups in the die manifold and land are plotted from an analytic calculation compared with those from anumerical computation using commercial software. These charts help an engineer to design a plastics diecasting sheet and film. The comparison between the analytic and numerical reveals that the analytic chartscan only be used for thin film die design, while the other can be used for a thicker sheet. A design exampleshows that the die shape built from our charts is in agreement with    previous design at the same deformationrate.  Keywords : Coat-hanger die, Plastic sheet Casting, Slit die, Die design 1   Introduction Plastics sheet productions have three important processing steps: molten plastics preparation, sheetformation, and solidification. The molten plastics isestablished in an extrusion machine managing the production capacity, then the plastics sheet is formed  by a slit die, and finally solidified in air or water  before entering a rolling system controlling the final product thickness. Thus, the slit die governs the plastics sheet approximated thickness and the final product quality. In general, the slit die is composed of two main sections [1, 2]: manifold and die land.The manifold  delivers the polymer melt across thedie width and the die land  forms the polymer melt to be a plastics sheet and performs the molten polymer flowing uniformly across the die lip. In some cases, achoker bar and flex lip [3, 4] are attached at the dieland end to equalize the molten-plastics flow acrossthe die width. The slit dies have three main designs:Tee, fish tail, and coat-hanger die (see Figure 1). TheTee die is the first and the simplest design that doesnot contribute the molten-plastics flow inside the dieacross the die width. Thus, the polymer melts flowvery fast at the middle and slow near the die edges.The fish tail and coat-hanger dies are the developed designs helping the flow more uniform across the dielip. However, the fish tail die provides a very lengthydie land and thus a cumbersome die. The coat-hanger die is the most versatile design. It can be designed tofit a compact production site. Figure 2 shows the coat-hanger die with its manifold and land without the choker bar. The manifold is acurved and round tube tapered from the middle to thedie edges. The manifold curvature controls the dieland length and eventually the total size of the die. Insome cases, the manifold cross section may bedesigned in trapezoidal, rectangular, triangular or eyedrop shape [5]. Sometimes, the manifold and land aredesigned such that the polymer melt flows in onlyhalf of the cavity, while the other half is flat for readily machining and cost saving [6]. The design of coat-hanger die is not only useful for thin sheet and film productions, but also for pipe manufacturing and      Arunworradirok S. and Kolitawong C. 36 © King Mongkut’s University of Technology North Bangkok Press, Bangkok, Thailand    blow molding [1, 2, 7]. In those processes, the polymer melt from the metering section in theextruder screw can be uniformly distributed to forman annular shape by using the coat-hanger diecenterpiece.(a)(b)(c) Figure 1 : shows the slit die designs: (a) Tee die,(b) fish tail die, and (c) coat-hanger dieA coat-hanger die design for a power law fluid was previously analyzed by many researchers [8, 9, 10,11]; however, the manifold curvature is very lengthyand design adjustment may be necessary due to in-line space limitation and cost reduction [12].To achieve the design adjustment, Smith, D. E. et  . al .[13, 14] apply a numerical optimization incorporatingwith 2-D Hele-Shaw flow simulation in the designwith minimum pressure drop and reduced velocityvariation across the die exit. The optimal die shape,however, is very complex and difficult tomanufacture. Later, they have developed their method by restrict the manifold curvature and land width, i. e. fix the in plane die shape, and optimizeonly the manifold and land heights [3, 15]. Lately, Na, S. Y., and Lee, T. [16], apply the optimizationtechnique incorporated with a 3-D model finiteelement method (FEM) for a power law liquid flowin a coat hanger die. However, this techniqueexpenses lot of computer machine running time sincetheir optimal design reaches at the 4 th iterations or more.  H  l   P O    R in   P 1    R O    L y x y ( l )    y O   α   s    R ( l ) Figure2 : shows the coat-hanger die with its manifold and land without a choker bar    Moreover, Sun, Y. and Gupta, M. [17, 18]numerically study the elongational viscosity effectsof a low-density polyethylene (LDPE) melt flowingin a coat-hanger die. The elongational viscosity isfound to slightly affect the velocity at the die exit.   Inaddition, Qu, J. and Zhang, X. [19] examine a coat-hanger die under axial vibration extrusion. Theyfound that the axial periodic pressure wave helpimprove the mechanical properties, such as tensile,impact strength and young’s modulus, in polypropylene sheet since the pressure pulsationincreases the polymer crystallinity.Michaeli, W. [1, 2] describes Wortberg, J. and Kirchner, H.’s coat-hanger design for a laminar,isothermal, and incompressible molten polymer flowin the die without end and edge effects and viscoelastic behaviors. They use a representativeshear rate γ    and viscosity η  on both the manifold and die land. The representative   shear    rate   γ    is aconstant value at which the non-Newtonian shear rateequals to the Newtonian one. Since the representativeshear rate correlates the representative viscosity, therepresentative viscosities in the manifold   R η  and inthe die land  S  η  are the nominal viscosities throughany cross section area of the manifold and die land,respectively. Figure 3 shows the representative shear rate conceptin a capillary. In the figure, the shear rate of a non- Newtonian fluid  ( ) non Newtonian γ   −  flowing in a capillary     A Slit Die Design for Casting Plastics Sheets© King Mongkut’s University of Technology North Bangkok Press, Bangkok, Thailand    37   τ  () τ η γ γ   = ⋅   0 S  r R e ≈ ⋅       S     r      R γ    .2 S  r dPdy τ  = ().2 dP r r dy τ  = .4 4()  Newt  V r r  R γ  π   =     y 4 4 S  V r  R γ  π   =    ()  Non Newtonian r  γ   −    Wall Newt  γ      Wall non Newt  γ   −  < V const  = s r Representative radius in the capillary =  Representative shear rate γ   =   Representative shear stress τ  =   Figure 3 : shows an idea of the representative shear rate in a capillary [1, 2].The shear rate of the non-Newtonian fluid flowing in a capillary radius  R matches that of the Newtonian one at the representative radius r  s .radius  R matches that of a Newtonian one ( )  Newt  γ    atthe representative radius r  s . Thus, the volumetricflow rate in the manifold at any given length l is, ( )  RO lVlV  L =   (1)   where O V   is a half of the total volumetric flow rate inthe coat hanger die; L , the half-die width; and  l , alocal coordinate measured from the die edge (see Figure2). When the shear rate along the manifold issteady, the representative shear rate of the polymer atthe middle of the manifold, ( ) O  R γ    , equals that at anymanifold radius,  ( )  R γ    . This gives, 13 ( ) O l R l R L  =   (2)where  R O is the maximum manifold radius located atthe middle of the die.   Thus, the total pressure dropacross the die is the combination of the pressure dropthrough the manifold and the die land which is, ( )( )( )( )( ) l y H  LV lds Ls RssV lP OS  Ll RO 34 128  η π η  +=∆ ∫ (3)   where  H  is the die lip thickness, or in other words,the approximate polymer sheet thickness. For auniform velocity across the die outlet, the pressuredrop in the die must constant along the die width.That is, 0)( =∂∆∂ lP (4)Then, Wortberg, J. and Kirchner, H. found that themanifold curvature  y ( l ) is, 32 )(     =  Ll yl y O (5)   where  y O , the maximum die land located at the diecenter, is 3 24  ROS O  H L y R η η π  = (6)   If the representative viscosities in the manifold   R η   and in the die land  S  η  are the same, then Eqs. (2) and (6) give, ( ) 123 0.482 O  R BH  = (7)   ( ) 123 1.474 O  y B H  = (8)   where the total die-lip width,  L B ⋅= 2 (9)       Arunworradirok S. and Kolitawong C. 38 © King Mongkut’s University of Technology North Bangkok Press, Bangkok, Thailand   Wortberg, J. and Kirchner, H. proposed a schemeshown in Figure 4 to design a coat-hanger die whichcan be used for the operating condition independenceon the left path. They examine the pressure dropacross the die by substitute l =  L in Eq. (3) to get, 3 12  H  L yV P OS O η   =∆ (10)   and the residential   time,  DOO t V  y H  Lt  <=  (11)   where t   D is the degradation time of the polymer melt.Moreover, Michaeli, W. also informs that Görmar correlates the flow rate O V   and the pressure dropacross the die P ∆ by two dimensionless sets; one for the flow in the manifold and another one for that inthe die land, Figure 4 : shows Wortberg and Kirchner’s flow chartto design a coat-hanger die [ Error! Bookmark notdefined. ]. Figure 5 : illustrates Görmar’s dimensionless chartsto design a coat-hanger slit die for Prandtl-Eyring’shyperbolic polymers [1].     ∆∆=  L R AP L R APC  RV  OOOO 31314 3 ϕ π   (12)       ∆∆=  AP y H  AP y H  LC  H V  OOO 21216 2 ψ   (13)   where ( ) u ϕ  and  ( ) r  ψ  are hyperbolic functionsdefined such that, ( ) ( ) −+−= 1cosh1sinh1cosh218 22 uuuuuuu ϕ  (14) ( ) −= r r r r r  sinh1cosh3 2 ψ  (15)       =  AC  τ γ   sinh  (16)  A and  C  in the Eqs. (12) and (13) are constants of thePrandtl-Eyring’s hyperbolic constitute equation,From Eq. (12)-(16), Görmar plots two dimensionlesscharts for a coat-hanger die design shown in Figure5. Since Görmar uses the Prandtl-Eyring’s hyperbolicconstitutive equation, his slit die design is only usefulfor the specific polymer. However, in general,engineers design a slit die for the shear thinning(power-law) plastics. Moreover, in many cases, thetotal die length is limited because of space restrictionand cost saving, thus redesign is a must. Here, weconcentrate ourselves on the steady, laminar,incompressible, power-law plastics flow in a coat-hanger slit die with a comfortable redesigned scheme.     A Slit Die Design for Casting Plastics Sheets© King Mongkut’s University of Technology North Bangkok Press, Bangkok, Thailand    39   Table 1 : shows the shear rates of the Newtonian and  power law fluids axially flowing in a round pipe and rectangular slit 2   Power law fluids Since a desired slit die must have a uniform pressuredrop along the die width, a proper and simple coat-hanger die design can help a design engineer to savetime on the various redesigns to fit the factory needs.Here, we model a coat hanger die for a steady,laminar, incompressible, power-law polymer flow ina slit die. Table 1 shows the shear rates of the Newtonian and power law fluids flowing in a round  pipe and slit [10, 20, 21]. For a round pipe of radius  R in Figure 3, the representative radius r  s at whichthe shear rate of the non-Newtonian fluid  ( ) non Newtonian γ   −  equals that of the Newtonian one  ( )  Newt  γ    is [22, 23], 1 1 134 nns O r Rn −   = +     (17)   and the representative height,  y s , in the slit is, 1 21312 −     += nns n H  y (18)   For a power law fluid, the viscosity is a function of the rate of deformation, 1 − = n k  γ  η   (19)   where k  , a consistency index, and  n , a power lawindex, are constants for a specific polymer and temperature. Thus, substitute the power law shear rate of a round pipe in Table 1 at the representativeradius r  s [Eq. (17)] into Eq. (19) to get therepresentative viscosity in the manifold, 13 314 −     ⋅⋅    +⋅= nOOn R  RV nk  π η   (20)   Similarly, the representative viscosity in the die land is, 12 2213 −     ⋅⋅    +⋅= nOnS   H  LV nk   η  (21)   Thus, one can determine the maximum die land   y O  for the power law fluid from Eq. (6). Moreover, fromEq. (3), one can integrate the pressure drop in the dieto get,             +    −=∆ 3233232431 1212  Ll y LH V l L R LV P OOS O RO  η π η  (22)From the pressure drop of a power-law fluid flowingin the die, Eq. (22), we propose dimensionlessrelations [24, 25, 26] between the pressure drop inthe die and the flow rate in the manifold as,             ⋅⋅⋅=    ⋅⋅∆⋅  L R LV  LP O RO R R ,21 22 π η  ρ ψ η  ρ   (23)   and similarly, in the die land,             ⋅⋅=    ⋅⋅∆⋅ OS OS S   y H  LV  H P ,21 22 η  ρ ψ η  ρ   (24)   Then, from Equation (22)-(24), we can plot relations,in the manifold, between the dimensionless pressuredrop across the die versus the dimensionless volumeflow rate for the dimensionless maximum manifold radius,  R O /  L , from 0.01 to 0.05, and at the same time,we plot those, in the die land, for the dimensionlessslit thickness,  H  /  y O , from 0.01 to 0.2 as shown inFigure 6.Here, we plot by using a typical HDPE at453 K (180 ° C) which has the consistency index, k  ,6,190  N  ⋅  s n / m 2 , the power law index, n , 0.56 and thedensity,  ρ  , 950 kg / m 3 [11]. Figure 7 shows a novelcoat-hanger die design schematic for the power lawfluid.  
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