Determination of an Extrusion Machine Performance Based on the Working Field of the Extruder Die

Some inventions along with theoretical and experimental research made it possible to increase the output of a thermally homogeneous melt provided by the screw. However, the quality of the extruded product depends on some specific features of the extrusion die and to a large extent on the rheological behavior (viscous and elastic) of the polymer melt. The mismatch between the design of the screwcylinder subassembly and the design of the extrusion die results in products with relatively short service life. The present paper has drawn up the working field of the extruder die and adjusted it based on the limitations imposed by the screw-cylinder subassembly, namely: the maximum output rate that ensures the required thermal homogeneity of the melt; the maximum output at which the heating system on the barrel (and possibly the screw) ensures the extrusion temperature; the minimum economic output corresponding to the diameter of the screw. The working field of some extrusion dies for blown films of the following polymers have been plotted: polypropylene, low density polyethylene, high density polyethylene and ethylene vinyl acetate.


Introduction
In their operation polymer processing machines thermally pollute the environment, which results, among other things, in relatively low energy efficiency of machines operating in a polytropic mode.
On the other hand, in the case of plastics extrusion, the mismatch between the design of the screwcylinder subassembly and the construction of the extrusion die results in products with internal stresses and inhomogeneous cross-section. The lifetime of these products is relatively short. In the case of pipes, for example, fracturing before the prescribed service life can lead to the environmental pollution caused by the substances flowing through.
Therefore, it is necessary to analyze the working field of the extrusion die considering the parameters of the extrusion process, the behavior of the polymer melt, the conditions imposed by the quality of the product and some economic considerations.
Extrusion machines have been improved particularly after 1953, when the first in-depth theoretical analyses of the extrusion process were published [1][2][3][4][5][6][7][8][9][10]. The main objective was to design and manufacture the extruder screw in such a way as to ensure the highest possible output of a thermally homogeneous melt.
There were created spiral threaded screw barrier and then consequently intensive mixing zones on the screw [11]. To increase the output, grooved zones were provided on the cylinder in the feed zone, which by increasing the friction coefficient between the polymer granules and the cylinder resulted in increased machine output [12].
Experimental and theoretical investigations in relation to the machine output, calculation relations established for the output [13; 14], design solutions and relations corresponding to zones of intense homogenization [15][16][17], as well as those related to the establishment of thermal inhomogeneity and melt temperature variation in the screw channel [18] led to the increase of the screw potential output.
To optimize the extrusion process, initially the machine optimization diagram was plotted upon output-pressure coordinates at the outlet of the extruder screw. Subsequently, the working field of the extruder die was defined and plotted on the output-pressure coordinates at the extruder die inlet. The ensuing argumentation puts forth an improved, completed version of how to define the working field of the extruder die.

Extruder die working field and micro working field
In the extruder die the melt pressure decreases from pe, at the inlet of the extruder die, to value pf in the end section of the die. Pressure pf may be greater than or equal to the atmospheric pressure, p0; this depends on the geometry of the extrusion die, flow velocity and the elastic properties of the melt. The melt temperature at the extrusion die is taken to be constant and equal to the extrusion temperature, Te, characteristic of each polymer.
The working field of the extrusion die (in the half-plane delimited by the Gm coordinates -flowrate or mass output -and pressure pe) is obtained inside the contour determined by the intersection of the following curves ( Figure 1 , corresponding to the maximum allowable melting speed through the die. The drawn curves delimit the working field (ABCDEA) of the extrusion die as shown in Figure 1.
The actual working field of the extrusion die (ABCDEA, Figure 1) must be adjusted to consider the following ( Figure 2): the extrusion die is attached to a cylinder-screw subassembly characterized by a certain value of the inner diameter of cylinder D (45; 63; 75; 90; 100 etc. mm), which corresponds to a certain minimum economic output Gm,min min (curve 6); the construction of the screw-cylinder subassembly ensures a certain maximum output from the viewpoint of the thermal homogeneity of the melt, Gm,om (curve 7); the heating system of the machine cylinder (and possibly of the screw) makes it possible to attain the extrusion temperature of the melt, Te, up to a certain maximum value of the flow, Gm,th (curve 8).  The working field adjusted to suit the limitations imposed by the screw-cylinder subassembly becomes ABCDEFGA in Figure 2. , as in Figure 4. They must cross the working field of the extrusion die ( Figure 5). Otherwise, another screw, or extruder with a different diameter D shall be used (as the case may be, larger or smaller diameters).  The output of the extruder is proportional to the bulk density of the granular material in the inlet area of the screw. The bulk density varies randomly between a minimum, min , ρ v , and a maximum value, max , As a result, at a certain speed, n, the output provided by the screw, Gm, (randomly) takes values between the curves corresponding to these two bulk densities ( Figure 6). Also, because the melt temperature randomly varies between

Influence of melt viscoelastic behavior on critical parameters
The shape and dimensions of the working field depend on the rheological behavior of the polymer melt flowing along the path from the extrusion die, at temperature Te, under the action of pressure pe.
In the final nozzle of the extrusion die, which ensures the shape and dimensions of the finished product, between the melt and the die wall a shear stress p τ is exerted on the wall, corresponding to a shear strain p γ  .
If they become equal to or greater than their critical values cr p τ τ  and cr p γ γ    , then non-uniformities appear in the product (elastic turbulence, shark skin, etc.) as a manifestation of the elastic behavior of the viscoelastic melt [19,20].
In the case of the non-Newtonian viscous behavior of the melt, given by the Ostwald -de Waele law, there is a nonlinear dependence between the shear stress, τ, and the shear strain γ  , wherein τ K and  are rheological constants at temperature Te, whose values for a given polymer depend on temperature and pressure.
In the die, the pressure keeps the melt compressed. Because the melt is viscoelastic, normal stresses σ accumulate in it. When leaving the die, in the pressure environment  In both cases, the duration of relaxation of the normal internal stresses of the melt is where w is the velocity of the product after leaving the die. As a result of the superposition of the effects of normal stresses, σ, and the shear stress on the wall, based on the principle of critical energy [49], the following interdependence was established [21]: where ( ) Consequently, to have a maximum value of it, it is necessary for the normal stresses to relax completely before leaving the die.
It is useful for The maximum allowable output through the die (curve 5 in Figure 2) is calculated with the relationship, n cr ad is a safety coefficient in relation to the critical condition at the die wall and Gcr, is the critical output through the final zone of die (7).
By using relationship (8) one can draw the curve 5 in Figures 1 and 2.

Some practical cases
As to establish the working field of the extrusion die, the results obtained in papers [23,24] have been used.
For three blown film dies with different geometry of the flow channels one has drawn the working fields [23][24][25] for the following polymers: polypropylene (PP) two types of high-density polyethylene (HDPE -1 and HDPE -2) ethylene vinyl acetate (EVA). The viscous constants ( τ K and υ) and the critical shear rate ( ) cr γ  of the polymers have been written in Table 1. The critical shear rate increases with temperature increase.
The viscous constants ( τ K and υ) and the critical shear rate ( ) cr γ  of the polymers have been written in Table 1. The critical shear rate increases with temperature increase. The calculated output of polypropylene blown film is from extrusion machines built in Romania. The execution project was made at Polytechnics University of Bucharest (Project advisor V.V. Jinescu) and implemented by Sibiu Mechanical Factory. In general, the pressure drop along the extrusion die depends on the geometry of the channel, on the rheological behavior of the polymer melt, on the nozzle entry effect and the elastic effects and the sliding of the melt on the wall.
The extrusion die in Figure Figure 8b and c. A gap increased from 0.55mm to 0.65mm determined a shift of the curve for the maximum allowable output to higher values, because according to relations (7) and (8) 2 max ,~f ad s G .
On the extrusion die in Figure 9a, provided with spider mandrel, a rather narrow working field was obtained when processing high density polyethylene (HDPE -2) type 2 (Figure 9c).
The processing of high-density polyethylene (HDPE -1) resulted in the working field in Figure 9, b, with a wider working field than in Figure 9c corresponding to HPDE -2.  When processing low density polyethylene (LDPE) on an extruder with 60 = D mm equipped with the extrusion die in Figure 10a, one obtained the working field in Figure 10b. In case of the extrusion die in Figure 11a, in processing low density polyethylene (LPDE) one obtained the relatively narrow working field in Figure 11b. In processing high density polyethylene (HPDE -1) one obtained the operation range in Figure 11c. When processing ethylene vinyl acetate (EVA) as a blown film on the extrusion die in Figure 12a, the working field shown in Figure 12b, was obtained.  From the analysis of the results obtained, represented in Figures 8 -12, one can identify the influence of the following parameters upon the shape and extension of the working field of the extrusion die: -nature of processed polymeric material, individualized by the viscous and elastic behavior of its melt -extreme processing temperatures ( ) max min ;T T ; -type of extrusion die -geometry of the channel through which the polymeric melt flows, especially in the final zone (nozzle), before leaving the extrusion die -restrictions imposed for product quality and attenuation, until disappearance, of some effects of the elastic behavior of the polymer melt.
The operating point, the intersection of the functional characteristic of the extrusion die at the processing temperature with the functional characteristic of the screw, depends essentially on the: -processing temperature -screw velocity.

Sequence of calculations for finalizing the melt path through the extrusion die
The practical solution of the problem of maximum output at an appropriate quality of the extruded product, especially for complex cross section profiles can be done, at present, in the following stages, which may improve, in some cases, the proposed calculation method: one designs of the melt path through the extrusion die (the die profile) one draws the operation diagram (Figure 1) of the extrusion die (zero order approximation) one traces the functional characteristics of the screw (Figure 4) and the functional characteristics of the extrusion die ( Figure 5) one establishes the effective output based on the intersection of the functional characteristics of the screw and the extrusion die ( Figure 5) one makes a detailed analysis (for example, by the finite element method, CAD etc.) of the geometry of the extrusion die channels through which the melt flows (improvement in the velocity distribution at the die exit by flow simulation as to achieve a uniform melt flow in all sections) one analyzes the heat transfer between the melt and the extrusion die walls and the evaluation of the temperature profile across the thickness of the melt layer in the channel one corrects (adjusts) the sequence of channels in the extrusion die one recalculates and retraces the working field of the extrusion die (first order approximation), the operating characteristics of the extrusion die and one determines the working point and consequently, the output if the difference between the outputs of the two approximations (zero and one) is relatively large, we move to the second order approximation and so on. In successive stages, the profile of the extrusion die flow zones is optimized.

Conclusions
The working field of the extrusion die was delimited in the output -pressure diagram of the extrusion die inlet (Gm -pe) by resorting to an analysis of the flow through the extrusion die, by considering the regime parameters, the rheological behavior of the polymer melts and the influence of its elastic behavior on the quality of the extruded product, The working field thus obtained was adjusted by considering the flow limitations imposed by the screw-cylinder subassembly regarding: -the minimum output corresponding to the screw diameter; -the