Analysis of pellet shaping kinetics at the die opening in underwater pelletizing processes.
There are currently several different ways of pelletizing thermoplastic polymers, which can be divided into two main groups: Strand pelletizing and die-face pelletizing. In the first case, the melt strands are cooled down, solidified and then cut. Die-face cutters cut the melt directly at the die opening, before a cooling medium, usually air or more often water, transports the freshly cut particles away, solidifying them in process . Because of its advantages in terms of throughput, automation, pellet quality and applicability to a large variety of thermoplastics, underwater pelletizing has gained high importance within the last years among the different pelletizing technologies . However, certain pellet defects may occur, if material and process characteristics and are not well adjusted. Deformation of the melt because of water flow at the die-hole opening may result in undesired shapes. Low viscous polymers are particularly sensitive to this effect. Other pellet defects are vacuoles, filaments, or agglomerations . An example is shown in Fig. 1. By today, the actual cutting kinetics of underwater pelletizing are widely unknown and only a small branch of research is addressing this topic.
The cutting process takes place within a contained water box at high cutting speeds. The shaping of the pellets is influenced by the rheological and thermodynamic characteristics of the polymer and specifically dependent on its viscosity and its elasticity. Convective cooling and the deforming drag force of the water flow also have an effect on pellet shape.
Of the very few existent studies on underwater pelletizing kinetics, most cover the flow behavior within the die, but rarely has any study looked at it the die opening. Dixit et al.  investigate the fluid dynamics within the die with a focus on pressure drop caused by flow channel narrowing. They develop a numeric mathematical model to calculate velocity, temperature, and pressure profiles within the die and find the latter to be highly dependent on viscosity and throughput. The risk of a die-hole freeze off and consequentially a wide pellet size distribution decreases with a higher pressure drop. Thus, more viscous materials are regarded as less susceptible towards that effect. This is in contrast to simulative findings by Henderson , which show a more narrow pellet size distribution (indicating constant specific throughput) when low viscosity additives are used. Henderson and Schon  also investigate the phenomenon of die-hole freeze off and its dependency on the temperature gradient from the die to the water in an experimental setup, using four different capillary designs. Their findings show an influence of capillary design and resulting shear rate on the likelihood of freeze-off effects.
Neubauer et al.  are the first to take a look at the effects at the die-hole exit, as they take the shear stress because of capillary flow as well as the stress acting on the melt during the cutting into account. Additionally, they use a Deborah number connecting the polymer's relaxation time to the processing time, in order to determine the brittleness of the material will behave when being cut. They deduce that low cutting distances (from the blade to the die plate), high cutting velocities and increasing throughput will cause a more brittle behavior, which they regard as beneficial for processing. However, the actual kinetics during the cutting and the influences of higher stresses on resulting polymer shapes are not taken into account.
This study aims to analytically describe the pellet shaping kinetics at the die-hole opening and to define a nondimensional number representing the equilibrium of forces at the pellet surface. This number can serve as a measure for the pellet's capability to withstand the deforming influences during the cutting and to preserve a spherical shape. Additionally, this study will verify stated theoretical concepts via an experimental research. This is done using a specially designed camera system which allows for an observation of the cutting process inside the water box.
ANALYTICAL DESCRIPTION OF PELLET SHAPING KINETICS AT THE DIE OPENING
The basic shaping process can be seen in Fig. 2. Between two cuts hot melt exits the die-hole, obtaining a spherical form. For simplification we assume that the pellet can be described as a perfect sphere segment with a radius R and a height h. The time frame of the shaping process is defined by the geometry of the die plate, which is shown in Fig. 3, and the circumferential velocity of the blades. The following calculations are exemplary for four die-holes and six blades with equidistant positioning. The blades run along a circular arc b with the radius [R.sub.h]:
b = [R.sub.h] x [pi]/3. (1)
The blade velocity [v.sub.b] can be calculated from the rotational speed [n.sub.d] of the drive shaft:
[v.sub.b] = 2 x [pi] x [n.sub.d] x [R.sub.h]. (2)
The time [t.sub.C] between two cuts can then be obtained from the ratio of the circular arc, where the blades run along, and the blade velocity:
[t.sub.C] = b/[v.sub.b] = (3)
The pellet growth is defined by the volume flow through the die. The volume of the spherical segment thus can be described as
V(t) = [??] x t = [pi]/3 x [h.sup.2](t) x [2 x R(t) - h(t)], (4)
with t [member of] [0; [t.sub.c]], The pellet surface A is defined as follows:
A(t) = 2 x [pi] x R(t) x h(t). (5)
The relation between radius and height is given by
R(t) = [a.sup.2] + [h.sup.2](t)/2 x h(t) (6)
with h being defined by a cubic equation:
[h.sup.3](t) + 3 x [a.sup.2] x h(t) - 6 x V/[pi] = 0. (7)
Applying Cardano's method leads to two imaginary and one real solution, the latter being
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
Based on the pellet volume from (4), an equivalent pellet radius [R.sub.p] can be calculated:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
Figure 4 presents the development of height h and radius R of the spherical segment, as well as pellet surface A and equivalent pellet radius [R.sub.p], each in dependence on pellet volume over time. Logically, a convergence of R and [R.sub.p] can be observed. Whereas the pellet surface A grows linearly, the segment height h shows a concave progression. Consequentially, the strain rate at the pellet surface decreases at higher pellet volumes.
Analysis of Pellet Shape Resistance Against Water Flow Drag
The geometrical considerations above apply to a perfect sphere. However, underwater pelletized granules usually show different grades of deviation from the spherical form. One reason for this is the deforming influence of the water flow force, which acts on the pellet during the shaping process, as the melt exits the die-hole. The extent of deformation is dependent on the equilibrium of forces on the pellet surface, which is determined by process parameters and material characteristics. For reasons of better legibility, time dependence of factors is not made explicit in the following equations. However, all geometrical factors describing the pellet and all factors dependent on those are dependent on time.
As the water is moved by the blades, flow speed at the die-hole opening can be approximated with the blade speed [v.sub.b]. The resulting drag force [F.sub.D] acting on the melt/pellet can, again assuming a spherical shape, be calculated  by
[F.sub.D] = [C.sub.D] * [rho]/2 x [v.sup.2.sub.b] [A.sub.proj], (10)
where [A.sub.proj] is the projection of the area hit by the water flow and [c.sub.D] is the nondimensional drag factor. As the Reynolds number for any realistic blade speed is far bigger than 2,380, turbulent flow can be assumed and [c.sub.D] can be set as 0.4 . [A.sub.proj] is given by
[A.sub.proj] = [R.sup.2] x [[pi]-arcsin a/R + 1/2 x sin (2 x arcsin a/R)] . (11)
Consequentially, the drag force induces shear stress [[tau].sub.A] along the cross section of the die-hole exit
[[tau].sub.A] = [F.sub.D]/[pi] x [a.sup.2] (12)
which leads to a line tension [T.sub.drag] along the pellet's surface:
[T.sub.drag] = [[tau].sub.A] x R. 03)
[T.sub.drag] therefore poses a deforming influence caused by the water flow drag.
Simultaneously, as the pellet grows it is subject to a constantly increasing strain because of volume and surface enlargement. Strain along the pellet's surface is equibiaxial and can be derived from the ratio of pellet circumference C and diehole diameter a:
[[epsilon].sub.C](t) = ln C/2 x a, (14)
C = 2 x R x ([pi] - arcsin a/R). (15)
As mentioned above, the strain rate is decreasing over time. The average strain rate between two cuts can be approximated by
[[??].sub.C] = [ln C/[2 x a]]/[t.sub.c]. (16)
The resulting tensile stress along the circumference can be calculated at any Azimuthal position from the strain rate and the extensional viscosity of the polymer. Applying Trouton's ratio, the extensional viscosity derives from the zero shear viscosity of the material multiplied by three, which leads to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)
In accordance to the calculation of [T.sub.drag], the normal stress resulting from surface strain can be converted into a line tension along the meridian [C.sub.m]:
[C.sub.m] = 2 x n x R, (18)
[T.sub.strain] = [[sigma].sub.C] x A x [C.sup.-1.sub.m] = [[sigma].sub.C] x h. (19)
[T.sub.Strain] can be regarded as an inner tension of the pellet, which poses resistance towards the attacking drag. Thus, the ratio of [T.sub.drag] and [T.sub.strain] serves as a measure of the pellet's deformation sensitivity (DS):
DS = [T.sub.drag]/[T.sub.strain]. (20)
The DS number can be used as a tool for successful underwater pelletizing process design. In dependence on material properties, volume flow rate, cutting speed, die-hole diameter and process water, DS can be calculated to anticipate pellet deformation and adjust parameters in order to achieve more spherical shapes.
VISUAL ANALYSIS OF THE CUTTING PROCESS
The experiments were carried out on a 0 60 x 30D single screw extruder by Gottfert, Buchen/Germany and an underwater pelletizer by BKG Granuliertechnik GmbH, Munster/Germany. The setup is shown in Fig. 5. The melt passes the bypass--which is installed for start-up and rinsing procedures--and enters the die plate leading into the water box. The die plate used in the experiments had 20 holes with diameters from 2.4 to 3.2 mm, all of which can be plugged and opened separately. The holes are positioned equidistantly on a circle with a diameter of 60 mm. Inside the water box, the melt exits the die-holes, forms out a spherical shape and is then cut off by the blades of a rotating cutter head. Via the cooling water, the pellets are transported to the separator, where they can be collected for analysis.
A camera system with a digital camera type "MARLIN F 046 B" by Allied Vision Technologies GmbH, Stadtroda/Germany and an 8 mm borescope by Karl Storz GmbH, Tuttlingen/Germany were installed for a visual analysis of the cutting process. As cutting frequencies in the water box are very high, the shutter speed of the camera has to be set within the range of 1-100 ps, which requires very bright lighting. Thus, two 300 W light sources with optical glass fibers were used, one conducting light through the borescope itself, the other one conducting it into the water box, so that both rays cross at the observed die-hole opening.
For the visual analysis a stroboscopic effect was exploited: the shutter of the camera is triggered through a controlling device, which can be synchronized with the blade speed. If stationary process conditions hold, each pellet shaping and cutting process will resemble the one before. Thus, synchronizing the shutter with the rotational speed of the cutter will lead to the impression of freeze-framing. By adjusting the time shift of the trigger, consecutive filming can be realized in order to analyze the cutting process, even at high cutting rates. With every rotation of the cutter, the shutter opens slightly later and images of several consecutive cutting processes can be combined to a seemingly continuous cutting process. If irregularities or instabilities are observed, the pelletizing process is nonstationary and needs to be adjusted.
The triggering was realized via a light curtain and a rotating disc with a conic slit. The disc was installed on the drive shaft of the cutter, as shown in Fig. 6. A displacement of the light curtain, orthogonally to the shaft axis, leads to a shift in the blade position captured. Therefore, a steady horizontal movement of the light curtain allows for an observation of the complete cutting process.
A variety of different thermoplastic materials with different rheological properties was used for this investigation, focusing on low viscous polymers. The selection contained amorphous as well as semicrystalline polymers (acrylonitrile butadiene styrene, two types of polystyrene and a polyamide 6.6). The main properties relevant for underwater pelletizing are summarized in Table 1. Extensional viscosity was calculated from zero shear viscosity, which had been measured on a rotational rheometer. Elasticity was evaluated by the calculation of reversible shear (following Laun , Wagner , and Winter ) and the resulting exit angles of the different materials. Melt temperatures were adjusted for each material (see reference temperature in Table 1) and then remained set.
In order to identify the main effects of different process parameters, the throughput, the rotational speed of the cutter head, the number of blades, the diameter of the die-holes and the water temperature were varied for the different materials. To guarantee comparability with industrial processes, which usually aim for constant specific pellet weight of 0.024 g, the parameter variation had to be adjusted accordingly. In some cases this led to a simultaneous variation of different process parameters, e.g. throughput and cutting speed. Details on the parameter variation can be found in Table 2.
For the analysis, the pellet shape and the size distribution were examined visually and via mechanical screening. It was found that size distribution is wider for a higher throughput, higher blade speed, a higher number of blades, and a larger die-hole diameter. An example of this is presented in Fig. 7. The same tendencies hold for influences on pellet shape: with increasing throughput, blade speed, blade number, and die-hole diameter, a higher deviation from a spherical shape occurs. Figure 8 shows an example for the influence of the blade speed. Additionally, an influence of water temperature can be observed, as higher water temperatures lead to more spherical pellets. These results show that the parameters, which Neubauer et al.  find beneficial for a more brittle material behavior and better processing, lead to undesired pellet shapes. Thus, there seems to be a trade-off between pellet quality and process efficiency.
It was also shown that low viscous polymers are a lot more sensitive to deforming through water flow drag. This can be seen in Fig. 9 which compares the cutting processes of polystyrene 158 K and polyamide 6.6. Even though the blade speed is slightly smaller for PA 6.6, the melt is distorted in flow direction, whereas the more viscous polystyrene preserves a more stable shape. Consequentially, PA 6.6 pellets are found to be highly deviating from a spherical shape.
Also, the PS 158 K pellets show a circumferential necking, which has yet to be explained. As this material shows the highest reversible shear (see Table 1), it can be assumed that this effect can be credited to the material's highly elastic properties.
In order to survey its adequacy and utility, the DS number was applied to the testing materials and parameters. The following results are exemplary for PS 158 K, a cutter head rotational speed of [n.sub.d] = 2604 [min.sup.-1] and six blades, the time between two cuts [t.sub.c] being 4 ms. Figure 10 shows the development of strain rate and DS number over pellet volume within the time frame [0; [t.sub.c]]. DS increases in V, as the projected plane exposed to the water flow is larger for higher pellet volumes, which agrees well with the experimental findings of higher throughputs leading to less spherical shapes. The decrease in strain rate for high volumes and the consequent decrease in the pellets' resistance against deformation also support this interpretation.
Furthermore, the less spherical shapes of pellets for which a larger die-hole diameter has been used, can also be explained via the DS number. Figure 11 shows the relation of die-hole diameter and DS number for all used materials, with larger diameters leading to a higher sensitivity to deformation. For this analysis, all pellets were evaluated at a state of comparable shape. This was done by determining the time at which a certain exit angle [alpha] (in this case a[alpha] = 41[degrees]) was reached and then calculating the DS number at this point. [alpha] can be calculated with the radii of the sphere and the die-hole. Thus, the DS number can be expressed as a function of the exit angle, the blade speed, the die-hole radius, and the extensional viscosity of the polymer:
DS = f ([alpha]; [v.sub.b]; [a.sub.;] [[micro].sup.-1]). (21)
It can also be seen in Fig. 11 how materials with a higher viscosity show a smaller sensitivity to deformation. Thus, high viscous materials allow for all deforming parameters, i.e. throughput, cutting speed and die-hole diameter, to obtain higher values, which is often required for reasons of better economic efficiency. The red area on the right indicates critical DS values of very low viscous polymers, for which no stable process conditions could be found in the stated experiments.
This paper shows that the interaction of material properties and process parameters have an influence on particle forming in underwater pelletizing processes. Different materials lead to different pellet shapes, even for identical process parameters. During the cutting process, high material viscosity leads to a better stability of the pellet against water drag. A nondimensional number was developed, describing the ratio of the tension induced by water drag and the tension because of material strain. This number can be used as an indicator of deformation sensitivity (DS) for a given material and a given set of process parameters. When designing underwater pelletizing processes for certain materials, this DS number may be a useful tool. In future works additional influences on the pellet shape should be included into the analytical model. Thermodynamic effects within the cooling track should be regarded in particular, as crystallization of the material as well as its heat capacity and conductivity may also be responsible for the final pellet shape.
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Oliver Kast, Kalman Geiger, Eberhard Grunschloss, Christian Bonten
Institut fur Kunststofftechnik, University of Stuttgart, Stuttgart, Germany
Correspondence to: Oliver Kast; e-mail: firstname.lastname@example.org
Results of the following investigation have partially been presented during the 20th Stuttgart Plastics Colloquium in 2007 at the University of Stuttgart, Germany.
Published online in Wiley Online Library (wileyonlinelibrary.com).
TABLE 1. Material properties. Material ABS Terluran GP22 Structure Amorphous Extensional viscosity 5,230,000 Pas (240[degrees]C) Reversible shear Angle 7.1[degrees] Material PS 158 K Structure Amorphous Extensional viscosity 269,700 Pas (200[degrees]C) Reversible shear Angle 22.4[degrees] Material PS VPT Structure Semi-crystalline Extensional viscosity 4,046 Pas (220[degrees]C) Reversible shear Angle 11.9[degrees] Material PA 6.6 Ultramid A3 Structure Semi-crystalline Extensional viscosity 454 Pas (280[degrees]C) Reversible shear Angle 3.8[degrees] TABLE 2. Parameters. Parameter Symbol [Unit] Varied parameters Throughput per die-hole [??] [kg/hl Drive shaft rotational speed [n.sub.d] [[min.sup.-1]] Number of blades [#.sub.b] Die-hole diameter a [mm] Set parameters Number of die-holes [#.sub.h] Cutting angle [[alpha].sub.b] in [degrees] Water temperature [T.sub.w] in [degrees]c Parameter Range Varied parameters Throughput per die-hole 5 to 25 Drive shaft rotational speed 1860; 2170; 2604; 3255 Number of blades 3; 4; 5; 6; 10 Die-hole diameter 2,4; 2,8; 3,2 Set parameters Number of die-holes 5 Cutting angle 90 Water temperature 50; 70; 80
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|Author:||Kast, Oliver; Geiger, Kalman; Grunschloss, Eberhard; Bonten, Christian|
|Publication:||Polymer Engineering and Science|
|Date:||May 1, 2015|
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