Polimery, 2002, Vol. 47, No. 6, p. 441449 Problems of engineering design and production of screws in counterrotating twinscrew extruders * J. Stasiek 1 and T. Nieszporek 2 Selected from International Polymer Science and Technology, 29, No. 9, 2002, reference PT 02/06/441; transl. serial no. 14841 Translation submitted by J.E. Barker INTRODUCTION The existence of a large number of designs of screw extruders is evidence of the complexity of the problems in extrusion of highmolecular materials. To increase extrusion productivity, improve product quality and especially for more effective plasticisation of difficult materials (including those that are thermally unstable), in addition to the singlescrew extruders that are still often used, various kinds of multiscrew extruders linear, planetary, cascade etc. have been developed [15]. Among the multiscrew extruders, the twinscrew extruders have found widest application. Both corotating and counterrotating twinscrew extruders are used; they have screws with flights that do not intermesh tightly, i.e. with some clearance between the flights, which has an important role in the plasticisation process. Corotating extruders characterised by highly effective mixing are commonly used in lines for production of polymer composites, whereas counterrotating extruders (the subject of the present article), owing to the slight dissipation of mechanical energy of the drive to the motion of the screws, are mainly used in the processing of PVC. Accordingly, in the present work we shall examine PVC exclusively as the polymer being processed. There are two basic geometric shapes of screws: cylindrical and conical. Figure 1 shows a modified design of a conical screw containing feed, preheating, mixing, compression, venting and metering zones [6]. In the final sections of the preheating zone and the middle sections of the mixing zone the flights have radial cutouts, called mixing recesses. This design, in which there is an appropriate variation of the reduction in volume of the channels along the zones, provides a much more favourable course of the process of final mixing of the material in the mixing zone [79]. The design of the screws must also ensure a suitable flow rate of the material in the individual zones of the screws, i.e. a greater transport capacity in the venting and metering zones than in the feed zone, to give incomplete filling of the channels of the venting zone with initially plasticised material. * Report presented at the VI Science Workshops "Processing of highmolecular materials". Poraj, 1922 June 2001 1 Metalchem Institute of Plastics Processing, 55 M. SklodowskaCurie Street, 87100 Torun 2 Czestochowa Polytechnic, Institute of Machine Technology and Production Automation, Al. Armii Krajowej 19c, 42200 Czestochowa Figure 1 Scheme of modified screw construction (A) and model course of compression degree changes (B); screw zones: a feeding, b initial heating and mixing, c compression, d degassing, e metering, M 1, M 2, M 3 mixing screw segments, f final polymer compounding zone with additives, H/h 1.5 T/69
GENERAL CHARACTERISATION OF THE DESIGN PROCESS FOR TWINSCREW PLASTICISING SYSTEMS The design process for counterrotating screw systems is dominated by experimental studies, aided by selected theoretical calculations, in contrast to the design of corotating systems, for which computer simulations are also used, relating to extrusion productivity and the pressure and temperature distribution in the material along the plasticising system. Scheme 1 shows the sequence of procedures for developing the design concept of a twinscrew plasticising system. The methods of concurrent engineering [10] occupy a special place in the modern approach to integrated processes of product development. Concurrent design generally means teamwork of the designers and technologists of the extrusion process, experts in evaluation of the physical properties of plastics and technologists specialising in plasticising systems [11]. Experimental studies carried out in simplified extrusion production lines are treated as modelling studies. Using the relations found there, assuming similarity in energy respects [12], it is possible to determine the basic design features of the proposed plasticising system and the conditions for extrusion of the material. One of the basic activities in elaboration of a design concept (of a new or modified plasticising system of an extruder) is selection of the interflight clearances in the individual zones of the screws, taking into account the manufacturing capabilities during screw manufacture. The aim of the present work was to analyse the flows between the flights and their influence on the plasticising process. In addition we present an analysis of the axial and frontal geometric profile of the screws, and the possibility of obtaining an appropriate value of the interflight clearances during machining of conical screws. Scheme 1 T/70
FLOW OF MATERIAL IN THE INTERMESHING REGION OF THE SCREWS One of the basic requirements for an extruder for processing thermoplastics is an appropriate ratio of extrusion efficiency to rotary speed of the screw, ensuring both a short residence time of the material in the plasticising system, and a low frequency of revolutions of the screw to limit wear of the plasticising system. Counterrotating twinscrew systems create besides the interflight clearances longitudinally and transversely closed segments of the screw channels: chambers. These ensure forced transport of the material, but they are also the cause of low efficiency of its lengthwise mixing. This efficiency can be improved by using appropriate designs of the zones of the screws (Figure 1) and especially of the compression or metering zone for example selection of suitable values of the interflight clearances in the individual zones of the screws. The value of the interflight clearances (gaps) (Figure 2) has a fundamental influence on the course of the plasticising process of PVC. The flows in the gaps counteract the flow of the material transported in the chambers, and at the same time make it possible for material to be exchanged between the chambers, thereby improving its homogenisation. The efficiency of the transport of the material, its intermixing and residence time in the plasticising system, as well as the stability of the process, largely depend on the value of the interflight gaps of the screws, i.e. on the intensity of movements of the material in the lateral and frontal gaps of the screws (Figure 3). The following types of flow of the plasticised material occur in the intermeshing region of the screws [4, 5, 13]: interscrew flow, interflight flow (divided into flow in the lateral and frontal, or cylindrical, gap) and leakage flow. Analysis of the strain rate of the material in counterrotating twinscrew plasticising systems shows that its maximum values occur in the frontal interflight gaps of intermeshing screws (Figure 4). One of the most important relative measures of the effectiveness of plasticisation, including homogenisation, is the work required to deform the material during extrusion. A measure of the work performed is the strain rate of the material, characterised by the transverse velocity gradient dv x /dy in the frontal gap of intermeshing screws. Approximate values of this gradient in counterrotating twinscrew systems can be calculated from the following relation: dv x /dy = (V 01 V 02 ) / S w (1) where V 01 and V 02 denote the circumferential velocity on the outside diameter of the screw and on the screw core (root diameter), respectively (Figure 3); S w is the frontal interflight gap (calender gap). This analysis of the twinscrew plasticisation process neglects the flows between the screws and the leakage flows, owing to their relatively low value compared with Figure 2 Axial crosssection of geometrical intermeshing elements: 1 right screw, 2 left screw; S bmin. side interscrew gap (minimal), S bmax side interscrew gap (maximal), S w calender gap, e screw crest width, b channel width, β flight flank inclination angle Figure 3 Mass transport in the isolated intermeshing gap: (a) screw crosssection; (b) longitudinal section of the intermeshing flight; D external screw diameter; h screw channel depth; V 01, V 02 circumferential speed on the screw outer diameter and screw core, respectively (assumed angular velocity ω = 1) the values of the interflight flows. Moreover, during interflight flows, especially cylindrical (Figure 4) there is at the same time, in addition to the distributive mixing process, an intensive process of shearing mixing, including a process of kneading and grinding of the plasticised material. Evaluation of the influence of the geometric elements of the screws, especially the dimensions of the interflight gaps (gap height and width), as well as the T/71
In the case of cylindrical screws, the volumetric flow rate of material through the lateral gap (V b ), which is the sum of the reverse flows (the drag flow and the pressure flow), can be determined from the simplified relation (onedimensional model) [14]: 1 3 πdn Vb = 2he pbsb S b (2) 12ηb cos φ Figure 4 Analysis of intermeshing flow in the calender gap: a) mass transport in the isolated intermeshing gap; V 01, V 02 circumferential speed on the screw outer diameter and screw core, respectively; S w calender gap; b) pressure (p) distribution; c) schematic presentation of forces acting on a particle within the highest polymer pressure area influence of the rotary speed of the screws (n) and the viscosity of the material (η) presents certain difficulties. Flows through the lateral gaps (i.e. occurring on both sides of the flight) merge in parallel with the flow through the cylindrical gap (Figure 5) and are caused by the same pressure drop, therefore they are closely correlated. Manufacture of screws with the required calender gaps does not come up against significant difficulties the outside surface of the flight crests is ground, and it is easy to obtain an appropriate shape at the bottom of the channels; therefore in the rest of this work we shall only analyse the flow through the lateral gaps of the screws. Figure 5 Fragment of the counterrotating twinscrew plasticising system showing the most important flow types: 1 right screw, 2 left screw, D screw outer diameter, h e intermeshing height of screws, n screw rotating speed, S b side interscrew gap, V b side gap flow rate, V c calender gap flow rate, V 2 leakage flow rate where S b denotes the lateral gaps, h e is the height of intermeshing of the screws, φ is the angle of inclination (lead angle) of the helical flights, n is the rotary speed of the screws, D is the screw diameter, p b is the pressure drop of the flow through the lateral gaps. As follows from relation (2), increase in h e, p b, S b, D and n causes a considerable increase in flow rate of the material. Excessively large values of V b (i.e. of the reverse flow) can cause outflow of material through the venting holes, and heat generated by friction can cause local thermal decomposition of the material. Both these effects, occurring separately or simultaneously, make it impossible for the extrusion process to proceed properly. PROBLEMS RELATING TO INTERACTION OF THE SCREWS Production of a plasticising system, especially a conical one, presents numerous difficulties. One of the complications is obtaining appropriate values of the lateral interflight gaps, including gaps with a constant value over the height of intermeshing of the screws h e in the metering zone. The values of the interflight gaps (lateral gaps and calender gaps) in the metering zone must be relatively small (< 1 mm), owing to the need for this zone to have a suitably high pressure, which is required in order to overcome the resistances to flow in the channels of the extruder head. The above assumption applies in particular to conical plasticising systems with a high material flow rate, where the end part of the metering zones is characterised by flights with a very steep slope of the helix. The screws of twinscrew extruders are generally machined by milling using a (taperedshank) end milling cutter with a rectilinear working surface, preferably by means of a disk cutter. The axis of the end milling cutter is perpendicular to the surface of the bottom of the channel and is displaced relative to the screw axis by an amount y. The required value of the displacement y of the milling cutter in the case of numericallycontrolled machine tools [15, 16] is found using calculation software that takes into account obtaining minimum deviation of the curvature of the axial contour of the screw flight from a straight line passing through the extreme points of the contour. It is defined on the basis of the value of the geometric elements characterising both the profile of the screw flights and the profile of the milling cutter (Figure 6). The maximum deviations of the curvature of the axial contour of the screw T/72
Figure 6 Scheme of relative positioning of the screw and mill during machining. Reference coordinates [17]: X s, Y s, Z s crew coordinates; X d Y d, Z d mill cutting coordinates; X 1, Y 1, Z 1 coordinates of the axial screw profile; d 1, d 2 screw diameters at beginning and end of the zone, respectively; h 1, h 2 axial screw profiles and end of the zone, respectively; β axial screw profile angle; δ mill cutter setup angle (perpendicular to notch bottom); r x vertical mill shift; x horizontal mill shift flight from the straight line, in the case of the theoretically calculated optimum displacement of the milling cutter y, are considerably less than a value of 0.1 mm [15, 17]. The distribution of the values of the lateral interflight gaps along the zone of the screws depends essentially on, among other things, the variation of the angle of inclination of the lateral surface of the flight (β) (also called the angle of inclination of the axial contour of the flight) (Figure 6). The value of the angle β (neglecting the deviation of the axial contour from a straight line) depends on the geometric elements of the screw and milling cutter and on y. This value can be determined from the relation [9]: φ + φ φ φ γ β = arc tg tg z w w z 2 2df sin sin y + 2 2 cos φz πdd ( 2h) h cos φzcos φw (3) where D is the outside diameter of the screw, h is the depth of the screw channel, d f is the minimum diameter of the end milling cutter, γ is the angle of the axial contour of the conical working surface of the milling cutter, y is the displacement of the axis of the milling cutter relative to the axis of the screw, Figure 7 Example of construction solution of the conical screw segment and axial profile of the slotting mill; screw zones: C degassing, D transient, E feeding, a and α parameters characterising flight milling φ z is the angle of inclination of the flight helix on the outside diameter, φ w is the angle of inclination of the flight helix on the root diameter. The angle γ has a considerable additional influence on the value of the angle β, especially in the case of large angles φ z. There is an additional, but less pronounced influence from y, on account of its low value (< 4 mm). Increase in diameter of the milling cutter (d f ) with simultaneous decrease in depth of the screw channel (h) causes the value of the angle β to decrease. It follows from analysis of equation (3) that the angle β of inclination of the lateral surface of the flight does not vary along the section of the screw where the diameter, values of the angles φ z and φ w and channel depth are constant. The above condition is fulfilled by cylindrical screws with constant pitch of the helix of the flight or flights. In the case of conical screws both with constant and with variable depth of the channels, in practical designs the angle β varies along the section of the screw; for example, when the pitch T (arising from the angles φ z and φ w ) has a constant value, angle β decreases considerably along the section of the screw owing to the increase in the angle of inclination of the flight helix. If during design of the screws there are variations of the angle β along the section, to obtain a constant value of the lateral interflight gaps of intermeshing screws there must be a corresponding variation in channel width (b) along the T/73
screw section. With increasing pitch of the flight helix (T) and decreasing channel depth (h) of the conical screw, there are marked changes in the values of angle β; they are corrected by changing the value of b along the section so as to obtain a constant lateral interflight gap (section E of the screw, Figure 7). In addition, when designing the screws of counterrotating twinscrew extruders it is also necessary to take into account the strength requirements of the geometric elements of the screws, mainly the width of the flight crest (e min = 5 mm), as well as the manufacturing capabilities, when the value of the diameter of the milling cutter is reached (d min = 6 mm). ANALYSIS OF THE AXIAL PROFILE OF CONICAL SCREWS MACHINED WITH A CONICAL SLOTTING MILL There is extensive literature on the machining of cylindrical screws used in worm gears [17], but discussion of other types of screws (for example hourglass screws or screws used in plastics processing extruders, especially in counterrotating twinscrew extruders) is rather sporadic. In fact these latter types of screws find quite wide practical application, and their technology is complex. Modern conical counterrotating twinscrew extruders also utilise socalled doubletaper screws [18], where the taper of the crests and the root taper (the taper at the bottom of the flight channels) of the screw have different values. In addition, the height of the flight profile (depth of the screw channels) and hence the screw profile (the angle and course of the axial profile of the screw) vary over the length of the screw. Precision manufacture of the screw requires continuous change of the relative setting of the screw and of the milling cutter (the distance between the cutter and the screw axis) during machining. Therefore screws of this type have to be made using numerically controlled (N.C.) machine tools [15, 16]. Appropriate control of the machine tool must be worked out beforehand on the basis of an analysis of the dependence of the variation of the axial profile of the screw on the geometric elements of the tool and on the parameters of the relative setting of the milling cutter and of the screw during machining. Setting of the milling cutter and the kinematics of machining of the screw During machining of the conical screw with a numerically controlled slotting mill with a rectilinear profile in the axial crosssection of the working surface, the axis of the milling cutter is set perpendicular to the generating line of the taper of the bottoms of the channels of the flights of the screw and at a defined distance from its axis. The distance of the cutter axis and of the screw axis (Figure 6) can be resolved into two components: the vertical displacement (r x ) depending on the movement of the cutter away from the axis of the screw in the radial direction for setting the cutter in the cut channel of the screw flights, the horizontal displacement ( y ) depending on the movement of the cutter from the axial plane in a direction perpendicular to the cutter axis and screw axis by a defined amount. For a nonzero value of y, the value of r x must be corrected so as to obtain the desired radius of the cut of its flights in a given crosssection of the screw. Correction of r x takes place in a plane parallel to the axial plane of the screw and at a distance from it equal to the horizontal displacement. The vertical displacement also varies over the length of the screw when machining conical screws. Therefore the setting of the milling cutter and of the screw depend on their relative mutual position, i.e. it must contain a kinematic parameter of the relative motion of the screw and of the cutter during machining. Furthermore, it is necessary to take account of the perpendicular setting of the cutter axis to the bottom of the channel of the screw flights at an angle δ (the semiangle of taper of the bottom of the channels of the screw flights). A system of equations describing the axial profile of the screw containing the vector equation of the generating line of the tool in the screw system and equations describing the envelope conditions [17] can be written in the general form: r s = r (, u φν, ) (4) s fu (, φν=, ) (5) 1 0 f2 (, u φν=, ) 0 (6) where u is a parameter of the position of a point on the straight line of the generatrix of the axial profile of the milling cutter; φ is a parameter of the working surface of the tool; and ν is a parameter of the family of the working surface of the milling cutter in the screw system. With reference to a specified value of parameter u, from the system of equations (5) and (6) we must find the corresponding values of the parameters φ and ν, so that on substituting the values of these three parameters in vector equation (4) we obtain the coordinates of the point of the axial profile of the screw. The cycle of calculations is repeated, each time changing the value of parameter u. By specifying values of this parameter in the range from zero to the upper limit (the number of points of the profile can be specified in the program) we find successive points of the axial profile of the screw. The angle of the axial profile of the screw is determined with reference to a straight line drawn through the extreme points of the axial profile (giving the position of the root and crest of the screw flight). The numbers of the screw flights are not used in the calculation program, but the width of the cut on the outside diameter of the screw (as an auxiliary quantity) is used for calculating the profile height of the screw flights. The width of the cut (kerf) is defined parallel to the screw axis. T/74
The user of the program for controlling machining alters the numerical data, taking into account the geometric elements of the screw being machined and of the milling cutter used. The calculations give the values of the angle of the axial profile of the screw (β) and the vertical displacement of the cutter (r x ), and the deviations of the axial profile of the screw from the straight line joining the extreme points of the profile. During calculations of different numerical values of y, using the computer program, it becomes possible to determine favourable values of r x and y, corresponding to minimum deviations of the axial profile of the screw from the straight line. Results of the calculations Investigations of the axial profile of the screw were carried out for a segment of the conical screw used in a modified 2T17/ 9M twinscrew extruder (designed at IPTS Metalchem Torun, and made at ZMCH Metalchem Gliwice). Figure 7 shows the design of the screw segment with a continuous flight, which is made up of three threeflight sections (C, F and D). The values of β and r x given in Table 1 confirm the variation of the course of the profile and of the profile angle in different crosssections along the axis of the screw. As follows from the calculations, there is a precisely defined value of the horizontal displacement of the milling cutter y corresponding to minimum values of the departures (deviations) of the profile of the screw channel from a straight line; in the case discussed above, for example, they do not exceed a value of 0.003 mm (Figure 8), whereas in the case of an unfavourable value, e.g. y = 0, the deviations of the profile exceed 0.3 mm. The right (pressure) and left sides of the profile of the screw flight channel must be machined separately, positioning the cutter on opposite sides of the axial plane of the screw. However, the horizontal displacement y (a parameter that is set on the machine tool) has the greatest influence on the profile angle and on the profile itself. If the deviations have to be minimal on the whole length of the screw, the parameters y and r x need to be varied continuously during machining. Therefore accurate machining of conical screws is only possible using numerically controlled machine tools. Within the individual sections there is linear variation of the horizontal displacement of the cutter, but as follows from Figure 9, the rate of variation is different along the individual sections of the screw. Table 1 Example calculations of axial screw profile angle ( β) and vertical displacement of the milling cutter ( r x ), assuming favourable values of y, for a conical screw in a modified twinscrew extruder Zone y, mm β r x, mm venting (C) start metering (E) end left profile right profile left profile right profile 2.125 2.103 3.500 3.517 9 57'26" 9 57'22" 9 58'53" 9 58'58" 33.151 33.156 23.170 23.170 Figure 8 Deviations of conical screw axial profile at the end of zone E on left flank ( y = 3.500), i.e. on the flight thrust side Figure 9 Advantageous horizontal mill shift ( y ) for individual screw zones; 1 left outline, 2 right outline T/75
Practical manufacture of a screw segment A screw segment containing sections C (venting), D (transition) and E (metering) was made using a machine tool from the company Waldrich (Coburg, Germany) with a slotting mill with minimum diameter d f min = 14 mm (γ = 10 ); its displacement y on the pressure side was 3.526, and on the opposite side of the flight it was 3.565. The constant value of the displacement y over the length of the individual sections of the screw segment arises from the technological limitations of the machine tool. A test for complete meshing of the screw segments (without chamfering the edge of the flights) was not possible in the final part of section E, where the pressure edges of the flight crests of one screw touched the lateral surfaces of the flights of the other screw (no lateral gap; the calender gap was approx. 4 mm). Figure 10 shows the form of the gap between intermeshing screws in the direction of rise of the flight helix. The minimum distances left between the lateral planes of the flights of the interacting screws vary along the height of meshing of the screws. On the diameter of the divisional taper of intermeshing screws, where the angles of deviation of the flight helices of the two screws are identical (φ 1st = φ 2st = approx. 39 ), the lateral clearance is maximum it is S st = approx. 1.9 mm, and is equal to the design value. On the other hand, on the outside diameter of the cone, on the pressure side at the edge of the flight crest the minimum lateral clearance is S b min = 0 (φ z = approx. 32 ), and the maximum lateral clearance S b max = 0.9 mm (φ w = approx. 51 ). Suitable chamfering of the edge of the flight crests (a x α, Figure 7) in the end part of section E eliminated this shortcoming, giving a minimum clearance greater than 0.6 mm. A preliminary analysis of the course of the frontal profile of intermeshing screws was carried out, taking into account the problems encountered in obtaining suitable lateral clearances (with a constant value over the height Figure 10 Fragment of the placing of conical screws in the feeding zone E: a) cross section, b) longitudinal section, c) longitudinal intermeshing gap towards screw helix inclination (minimal distances between interacting screw fights); S st flight gap on intermesh diameter Figure 11 Frontal profile of the intermeshing conical screws at end of zone E of the screw segment: 1 right screws, 2 left screw, S lateral gap in front crosssection of meshing of the screws h e ) in a plane perpendicular to the line of inclination of the flight on the partition diameter of the cone of the screws [19]. Figure 11 shows the frontal profile of intermeshing screws at the end of section E (assuming S w = 0); it follows from this that there is no clearance on the outside diameter of the cone (the clearance has a negative value of 3.1 mm), whereas it already has a positive value (1.3 mm) on the partition diameter. Analysis of the frontal profiles of intermeshing screws in the other sections (C and D) of the segment confirms the measured results obtained, relating to lack of constancy of the values of the lateral clearances along h e, when a rectilinear axial profile of the screw channel is assumed. CONCLUSIONS One of the main activities when elaborating the design concept of a new or modified plasticising system of an extruder is selection of the interflight clearances in the individual sections of the screws, taking into account the manufacturing capabilities (technological limitations). Analysis of the axial profile of the screws confirms that its course varies in different crosssections along the screw axis. The investigations must take into account automatic searching for a favourable value of y by the program that is elaborated (by the method of successive approximations). The results of evaluation of the frontal profiles of intermeshing screws confirm lack of constancy of the values of the lateral clearances over the height of meshing of the screws, if it is assumed that the profile of the lateral surface of the screw flight in axial crosssection is approximately rectilinear. If, during design of the screws, the angle b is varied along the section, then in order to obtain a constant value of the lateral interflight clearances of intermeshing screws (Figure 2) there must be a corresponding variation in width of the channels (b) along the section. Moreover, in sections where the values of the angles of inclination of the helix on the partition cone φ st are greater than 35, and the T/76
difference between the angles of inclination (φ w φ z ) > 15, it becomes necessary to provide appropriate chamfering of the edge of the flights, or an increase in the lateral clearance. There is an urgent need to carry out research into the interaction of screws, including screws with an involute frontal profile [19]. The clearances between interacting screws are different in different places of the flight surface. Thus, they are greater in the axial crosssection than in the frontal crosssection, and in turn are greater than in a plane perpendicular to the line of inclination of the flight on the partition diameter of the cone of the screws. The lateral interflight gap (Figure 10) should approximately have the form of a parallelogram in the case of cylindrical screws, as shown in Figure 2, or the form of a trapezium in the case of conical screws; the required difference in lateral clearances (S b max S b min ) is 0.10.3 mm. It is assumed that an approximately identical shear rate should act on material flowing through the gap along its width (h e ). REFERENCES 1. F. Hensen, W. Knappe and H. Potente: Manual of Plastics Extrusion Technology, I Principles. Carl Hanser Verlag, Munich Vienna 1989, p. 298 316. 2. J. Stasiek: Polimery, 1997, 42, 14. 3. The twinscrew extruder in the extrusion process. VDI Verlag GmbH, Düsseldorf 1991, p. 73123. 4. R. Sikora: Processing of highmolecular materials. Wydawnictwo Edukacyjne, Warsaw 1993, p. 5869. 5. J.L. White: Twinscrew extrusion. Carl Hanser Verlag, Munich Vienna New York 1990, p. 148191. 6. Polish Patent 179 289 (1996). 7. J. Stasiek: Plaste Kautsch., 1994, 36, 392. 8. J. Stasiek: Polimery 1995, 40, 542. 9. J. Stasiek: Plasty Kaucuk, 1998, 35, 70. 10. R. Rohaty ski: Proceedings of International Seminar on Tools and Methods of Concurrent Engineering TMC 96. Budapest 1996, Proc., p. 157175. 11. J. Stasiek: J. Eng. Design, 2000, 11, 133. 12. H. Langhorst: Practical design of screws for highperformance extruders. Thesis, RWTH Aachen 1989, p. 51101. 13. L.P.B.M. Jansen: Twinscrew Extrusion. Elsevier Scientific Publishing Company, Amsterdam Oxford New York 1978. 14. L. Fiedler, A. Pipiale and S. Marinow: Plaste Kautsch., 1994, 41, 191. 15. Information from the company Werkzeugmaschinenfabrik Adolf Waldrich Coburg GmbH & Co. (Germany). 16. Information from the company Weingärtner Maschinenbau GmbH (Austria): Screw manufacture, from design to documentation. Plast. Spec., 2000, 72, 16, 60. 17. T. Nieszporek: Mechanik, 1999, 72, No. 1. 18. Polish Patent 144 692 (1985). 19. P. Boral: Geometric analysis of a system of two conical screws with constant and variable pitch used in twinscrew extruders for transporting plastics. Doctorate thesis, Czestochowa Polytechnic, Czestochowa 2001. (No date given) T/77