I
SlLVlO ECCHER and ALDO VALENTINOTTI Pirelli Societd per Azioni, Milan, Italy
Experimental Determination of Velocity Profiles in an Extruder Screw This experimental verification of Newtonian flow theory in the special geometry of an extruder screw will be especially valuable for a clear understanding of what goes on in the channel of a rubber or a melt extruder
D E S I G N E R S , technologists, and scientists have devoted more and more time and effort to the study of screw extruders and the principles governing their performance, because the use of extruders in the production of rubber and plastic goods has acquired an ever-growing importance. A screw extruder is chiefly composed of a cylindrical sleeve, in which a special type of screw can rotate. Attached to the sleeve are a feed port and a head with a die at its end. The screw rotates in such a way that the fluid material adhering to the inner surface of the sleeve is dragged toward the head. The advancing material, having filled the head, is pushed against the die, whose resistance to outflow causes the extruder to fill and the pressure to rise. The pressure difference between the head and the outside causes the extrusion of the material through the die openings which give to the extruded material the desired shape and dimensions. Generally it can be assumed that the material does not slip a t the walls; this is especially true for wetting liquids or melts in any pressure condition and for the majority of rubber compounds when under pressure. In fact, with such high viscosity materials the slip a t the walls is prevented by friction forces acting on the surface of the screw and of the sleeve; these forces are proportional to the pressure built up in the material. Therefore, when the extruder is full the layer adhering to the wall of the sleeve does not move, while the layer adhering to the screw rotates with the same angular velocity as the latter. The intermediate layers are dragged by the rotation of the screw, and their velocity is lower in proportion to their distance from it. The movement of the fluid is made more complicated by the helicoidal axis of the groove and the shape of its section. T h e movement is always characterized by a laminar flow caused by viscosity forces. The extruder can be considered,
therefore, as a viscosity pump, whose performance is highly different from that of a centrifugal pump, which acts through inertia forces, and also from that of a constant-displacement pump. Hence, the investigation of extruders is primarily a rheological study and involves a knowledge of the rheological properties of the material used. This knowledge becomes of paramount importance when quantitative experimental results obtained with materials like rubber, which are highly non-Newtonian, have to be correlated (4); for nearly Newtonian materials, such as many plastic melts, the discussion can be limited to the laminar movement of a Newtonian fluid. With this assumption a mathematical solution can be attempted. Several attempts in this direction have been made by other workers based on further assumptions: The section of the groove is assumed to be rectangular, the axis of the groove rectilinear instead of helicoidal, the unavoidable clearance between the screw and the sleeve is neglected or its influence treated separately, and only movements parallel to the axis of the groove are considered.
In spite of the assumptions, mathematical derivations have given a t least a qualitative picture of the movement of the fluid inside the extruder and, for nearly Newtonian materials, also a rather accurate quantitative picture. They have shown the distribution of velocities in the groove, emphasizing the existence of a flow in the reverse direction to the forward flow; this is the only explanation for certain special phenomena in the extruder. To confirm these theoretical results, direct optical measuremehts of local velocity in three dimensions have been performed by using a transparent extruder sleeve and a transparent viscous fluid. The fluid used has almost Newtonian properties in the range of velocities of the experiment. Historical Much experimental work on the whole performance of the extruder has been published, including the studies by Rowel1 and Finlayson (73, 7 4 , Pigott (70)) McKelvey ( 7 ) , and Carley ( 2 ) . On the other hand, very few have studied
HEAD CUAMBER \ FEED PORT
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Qzzzzzw Section of typical screw extruder VOL. 50, NO. 5
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Experimental extruder and optical system Glass sleeve internal diameter 2 3 mm. Coaxial screw, diameter 22.6 mm., lead 30.4 mm., thread trough 15 mm. wide; thread land 15.4 mm. wide: axial section of groove 3.5 X 15 mm. (52.5 sq. mm.; perpendicular section 3.5 X 13.5 mm. ( 4 7 sq. mm.) C. Extruder head D. Feed port E. Variables orifice F. Microscope G. Dials H. Transparent chamber Castor oil, refractive index = 1.477 I. 1. Polyisobutylene mixed with paraffin oil
A. 6.
Here the apparatus i s assembled, and the transparent chamber i s readily visibfe a t the center
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INDUSTRIAL AND ENGINEERING CHEMISTRY
experimentally the distribution of velocities in the groove of the screw, and the numerous publications on this subject are mostly of a theoretical and mathematical nature (3). In 1868, Boussinesq (7) calculated solutions of the Navier-Stokes equations for the motion of viscous fluids in a rectangular groove with all four walls fixed. Rowel1 and Finlayson (73) solved the general equations for a rectangular groove with three fixed walls and one in motion, closer to the actual working conditions of an extruder. Later the same authors ( 7 4 , neglecting the effects of side walls, calculated and drew representative curves of the velocity vectors in the direction of the groove of the screw for the extreme conditions (maximum flow, no output) and for several intermediate ones. Rogowsky (Rigbi) (72) , giving weight also to the effect of the side walls, published isovels (lines joining all points having the same velocity) for both extreme conditions. The isovels corresponding to the no-output condition cannot be considered accurate. Later Grant and TValker (5) drew the profile of isovels for the no-output condition in a more reliable way; these results lvere confirmed by Carley and Strub (3). Experimental investigation of velocity distributions in screws has considerably lagged behind these important theoretical contributions. Tests with materials of different colors to investigate the nature of the motion and the flow lines can be considered the first experimental attempt to investigate the motion of the material inside an extruder. While studying the outflow of rubber compounds in his plastometer, Marzetti (9) had noted a telescopic motion of concentric layers. finding a disagreement with Poiseuille’s law. He used materials in two different colors, arranged alternately inside the apparatus. The experiment was repeated by Ricca ( 6 ) , who gave photographic evidence; previously Tritton (75) presented some pictures of an extruded rod; other workers tested differently colored compounds in a screw extruder. Grant and Walker (5)proved the existence of rotational motions in the groove by using color contrasts. The present authors also made similar experiments using a 2-inch extruder and a cylindrical orifice. The extruder was fed continuously introducing four rubber compounds with different colors one after the other. The extruded product was continuously covered with the first compound, and the section showed the concentric disposition of the other colors with the last compound in the center, while the thickness of the layer of the first compound was decreasing continuously during the extrusion. An orderly disposition of differently colored layers was also found in the groove of the screw.
SCREW VELOCITY PROFILES All these experiments demonstrate that there is a dragging motion of adjacent layers. Therefore it is possible to assume with certainty a laminar flow and to consider as zero or a t least negligible the slip along the walls. This is true not only for rubber compounds but also for plastic melts. Rigbi (11) reported an experiment claiming to substantiate the hypothesis of a back flow in the groove and thus proving the existence of an intermediate stationary layer. The material used was poly(viny1 chloride) ; this decomposes and blackens when kept at high temperatures for a sufficiently long period of time. The existence, at the end of the test, of an intermediate black layer was interpreted as proof of the existence of a stationary layer. Maillefer (8) has constructed the distribution of velocities in a rectangular groove in the two extreme conditions (maximum flow and no output). He used a rectilinear groove, with the upper wall movable, completely filled with viscous polyethylene. After injecting a series of colored streams in a section perpendicular to the axis of the groove, he moved the upper wall parallel to the axis. He then solidified the material by cooling it and reconstructed approximately the profile of the motion by cutting through the solidified mass along the colored lines. This is an ideal case and rather removed from the actual conditions in an extruder screw.
Detail of screw as viewed through transparent chamber
Experimental
For the experimental work a small, specially built extruder was used (diagrams and photographs). Essentially this was composed of a cylindrical tubular sleeje, A, mbde of transparent glass (internal diameter 23 mm.) and containing a coaxial screw, B, with only one rectangular flight. The screw was hollow to provide a return path for the forward flow of the extruder; this allowed the fluid to return continuously from the extruder head, C, to the feed port, D. A variable orifice, E, was used to control the outflow resistance of the liquid through the head and thus to vary the discharge rate. This device allowed the working conditions to vary continuously from maximum flow to no output. T o observe the flow in the groove, the screw was fixed, while the external glass tube was rotated; of course, this does not vary the relative velocities of the sleeve and the screw and therefore the velocity distribution of the fluid in the screw. Polyisobutylene mixed with paraffin oil and containing small particles of aluminum was used as the fluid for the microscopic observations. The observations were performed in a section of the groove (6-6, Figure 1) containing the plane passing through the axis of the
Figure 1. Perspective representation of velocity vector V and its components (tangential V t and binormal V,) at a point 0 in the section of the groove at a height h VOL. 50, NO. 5
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d q 5 mm
Y
-a.
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d-o,5 Vt = i
d.3,5 vt
=;
=
mm
1
mm
d-15
J
J d.75 c
Vt
=
mm i
1
b.
6.
Figure 2. Profiles of tangential components V , with depth of groove at three different distances d from one side wall a.
b. c.
Maximum flow Intermediate flow No output
screw, in three different working conditions : maximum flow, intermediate flow, and no output. A microscope, F, was placed with the optical axis perpendicular to the axis of the screw and in the plane of the section under study. The microscope had two micrometrical movements, one in the direction of the optical axis, which was the direction of the depth of the groove, and the other parallel to the axis of the screw for the transverse observation of the groove. Two dials, G, graduated to read to the nearest 0.01
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mm., measured these displacements and determined, through the position of the microscope, the coordinates of the focused particle; this was dragged by the fluid in which it was suspended and moved with the same velocity and local direction. T o correct the fictitious shortening of the distances due to refraction, measurements of the depths were multiplied by 1.5: This figure was obtained from the ratio of the actual depth of the groove and the corresponding distance read on the micrometer of the microscope,
INDUSTRIAL AND ENGINEERING CHEMISTRY
focusing the bottom and the top of the groove. The direction of the particle was obtained by orienting the ocular micrometer in the direction of the movement and measuring the angle with a goniometer attached to the ocular. The value of the velocity was determined by following the displacement of the particles against the scale of the ocular micrometer and measuring the corrrsponding time with a stop watch. T o minimize optical and mechanical errors, an accurate setup and testing were necessary. The tubular sleeve, made of special tempered glass without any optical anisotropy as shown by observation in polarized light, was ground on the inside to make this surface perfectly cylindrical. The glass tube was enclosed in a rectangular transparent chamber, H, filled with castor oil, I, to reduce refraction errors due to the cylindrical shape of the glass wall. This chamber, which in the lower part had a sealing gasket against the glass tube, was kept still. Preliminary testing was performed with and without castor oil in the chamber. Measurement of particles of known dimensions, suspended in the fluid, L. showed the need for this device. Careful consideration was given to ensure that the screw and the sleeve were coaxial and to maintain a constant clearance between screw and sleeve during rotation. This was obtained mechanically and was controlled optically. The optical equipment was chosen having in mind the sensitivity of the focusing. which led to a shortening of the focal length, without jeopardizing the possibility of observing all the depth of the groove. Hence a relatively high enlargement (70 times) was used. Rotational speed was therefore reduced to allow measurement of direction and .velocity without having time intervals too short. A velocity of 0.146 r.p.m. was used. This rotational speed might appear too low and therefore outside the actual working conditions of a rubber extruder; this fact is not relevant to the purpose of this work. Errors due to optical aberrations and to the curvature of the paths of the particles under observation were kept within negligible limits, reducing the area of observation to about one forth of the field of the microscope; this area was 1 mm. long in the generating line of the screw and 0.75 mm. (maximum) in the perpendicular direction. The error of a single measurement due to the focusing of the microscope was less than 0.1 mm., as proved by several measurements of the bottom of the groove. This was used afterward as a reference point; the standard deviation of a series of 30 single measurements was 0.047 mm. The fluid used in the experiment had a viscosity of 1530 poises at 20' C., a
SCREW VELOCITY P R Q F I L E I specific gravity of 0.852, and a refractive index of 1.495. The particles of aluminum filings had an average dimension of about 0.03 mm. Because of the high viscosity of the fluid and in spite of the higher specific weight, the suspended particles did not show any appreciable sign of falling in the stationary liquid.
Results and Discussion In elaborating the results, the section of the groove (15 X 3.5 mm.) was considered divided by a uniform system of lines parallel to the side walls (at a distance of 1 mm.) and of lines parallel to the bottom wall (at a distance of 0.25 mm.). Measurements of depth and of distance from the side were used to locate every result in the right position in the reticulum. An average of five measurements of velocity in each rectangle were made. The velocity vector, V , measured has been broken down into two components (Figure 1): V t in the direction of the tangent and Vb in the direction of the line binormal to the helix passing through point 0 under study, with the helix having the same axis and the same lead as the thread. The three vectors, V, Vt, and Vb, lie in a plane tangential to the helix a t point 0 (a-a, Figure 1). In fact, the third component, V,, normal to the helix is negligible in the central part of the groove and becomes noticeable only close to the side walls. The values of the two components V , and Vb are: Vt = vcos (CY P)
vb =
Vsin ( a
Only these "specific" velocities are considered below. The ratio between the components, Vo,and VO,,at the internal surface of the glass tube where p = 23' is:
Vob = sin - = 23" Vo; COS 23"
0.42
The average value of components Vt and V , enclosed in each rectangle was assumed to be the average velocity for the rectangle applied to its barycenter. Velocity Component Tangential to Helix. Variations of the tangential component of the specific velocity with the depth of the groove at three different distances from one of the side walls and in three working conditions (maximum Row, intermediate flow, and no output), respectively, are shown in Figure 2. In the first case (Figure 2, u ) maximum flow was not perfectly obtained, because of a slight back-pressure from the resistance encountered by the fluid on its back path. In the second case (Figure 2, b), with intermediate flow, the profile of the curve is almost parabolic: A zone of forward flow and a small back flow zone are apparent; hence the negative area of the profile is smaller than the positive one, agreeing with a flow rate in the forward direction. In this case graphical interpolation of the mean experimental points is shown on an enlarged scale (Figure 3) to give an idea of the degree of precision obtained.
In the third case, with no output (Figure 2, c), the profile of the curve i s similar to the previous case. Points A with zero velocity have risen still more, so that the negative area is almost the same as the positive area. The condition of zero output rate is fulfilled in the groove on the whole. A better representation for the whole flow in the groove is obtained by joining with lines (isovels) all points having the same velocity. The isovels are drawn in the axial plane section (b-6, Figure l), which is the section of the groove passing through the axis of the screw and was also the section of observation. Isovels from the experimental results (Figure 4) therefore show the velocities in the direction of the groove related to this section. At maximum flow (Figure 4, a) the isovels have a simple profile; the velocity is zero at the fixed walls and has the maximum positive value at the dragging surface, which is the internal wall of the glass tube. This value has been taken as unity in accordance with the definition of specific velocity. The velocity distribution is almost symmetrical; nevertheless, a slight thinning of the isovels occurs toward the right-i.e., toward the feed port. The tangential flow rate, obtained by integrating the tangential velocity, V,, over the whole section perpendicular to the median helix of the groove, was about 150 cu. mm. per minute, corre-
+ + P)
where CY is the angle formed by the direction of the path of the particle and a plane perpendicular to the rotational axis. /3 is the angle formed by the tangent to the helix of the groove and a plane perpendicular to the rotational axis (this angle varies from 23' a t the external helix of the screw to 29' a t the internal helix). CY /3 is therefore the angle between the direction of the particle and the direction of the groove, at the height of observation, h, measured from the bottom. The components in both directions of the velocity, VO,of the internal surface of the glass tube have been used as a standard for each ~3value: Vot = vo cos p
+
VOb = VOsin P The velocity components, v, and vb, of the particle have been related to them:
-vt--- V Vot
vo
cos ( a cos
+P P )
Figure 3. Graphical interpolation of mean experimental values corresponding to intermediate flow in center portion of groove See Figure 2,
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Figure 4. Profiles of isovels a.
Maximum flow
b.
Intermediate tlow No output
c.
sponding to an average velocity of about 3.2 mm. per minute. Figure 4. b, shows isovels in the condition of intermediate flow; the groove section is divided into two zones with positive and negative velocities The flow rate of the extruder, resulting from the differences in the two flows (positive and negative). was about 100 cu. mm. per minute. Any other value of flow rate could have been chosen by varying the opening of the orifice. ll'ith no output the zero line (Figure 4, c) dividing the two zones is moved upward from the bottom to about two thirds of the height of the groove. and the area with negative velocities is about twice the positive one. The highest negative velocity- is observed halfway 834
betbyeen the bottom and the zero Iine, and its value is slightly less than -0.3. -4 certain dissymetry of the isovels appears in the negative zone. In all three cases the shape of the isovels close to the side walls of the groove is of interest. The zero line rises and the upward bending of the isovels decreases as the condition of no output is approached. The effect of the walls is less noticeable as the ratio of height to width of the positive zone becomes smaller, The stationary layer acts for the positive flow almost as a new bottom of the groove. The positive flows in the three conditions can be considered as maximum flows in grooves with different hcight. As an example. the isovel $0.5
INDUSTRIAL AND ENGINEERING CHEMISTRY
in the three cases dirides, almost in the same ratio, the distances between the zero line and the upper surface of the groove. Velocity Component in Binormal Direction to Helix. The previous dis-
cussion concerned the velocity component tangential to the helix of the groove representing the real flow. The second component, perpendicular to the first one, does not contribute to this real flow. The vectors, representing the binormal velocities, have been shown in the axial plane secrion of the groove containing the points of their application. The vectors are therefore shown rotated by an angle fi. Figure 5 shows the detailed variation in the profile of the binormal velocity component, Vb,
SCREW V E L O C I T Y P R O F I L E S with the depth, a t ,three different distances from one of the side walls; Figure 6 shows the complete variation for the whole section of the groove. This distribution appears to be the same for all three cases. The movement is then not influenced by the tangential motion and can be considered as.independent from it. The principle of the summation of effects is here therefore valid. This binormal component from the maximum positive value obtained at the dragging surface, and used as unity, decreases to zero close to two thirds of the height of the groove (point B ) and increases in the other sense to a maximum negative value, which is slightly lower than -0.3; this value obviously goes to zero at the bottom wall. The profile of this component is therefore similar to that of the tangential component in the no output copdition (Figure 2, c; d = 7.5 mm.); hence in this case for both the components the positions of the zero velocity points ( A and B) are coincident. This agreement corresponds to a practical similarity of conditions-no output in both cases. In the case of the tangential component, the no output condition is caused by closing the orifice, while for the binormal component, excluding the leakage in the clearance between the screw and the sleeve, it is caused by the walls of the groove. The coincidence of the positions of zero velocity for both components in the condition of no output causes a stationary layer (point A =B, Figure 7, c). In all other working conditions, while the transverse distribution remains the same, the zero line of the tangential component is shifted downward. In this way two separate positions of zero velocity are created, where there is always either a mixing motion or a flow motion. The binormal component decreases in absolute value approaching the side walls of the groove, while the normal component, V,, tends to become more noticeable. This is parallel to the wails of the groove and has opposite signs a t the two ends of the groove. The circulation in the section is therefore complete. Comparison of the distribution in the middle portion of the groove and in the boundary portion shows that in the latter the flow rate is smaller (Figures 5 and 6). Total motion in the extreme section is altered by the motion parallel to the side walls, Quantitative measurements were not made as focusing was necessary; the visual observation nevertheless showed the existence of this motion in opposite directions a t the two extremes of the groove. I t was not possible to get reliable data on the leakage flow in the clearance
Figure z 5. ~~~
Profiles
Of
binormal component
v b
with depth of groove a i three different distances d from one side wall Valid for all three working conditions
Figure 6. Profiles of binormal component v b relative to plane section of groove passing through axis of screw Vectors are shown rotated through on angle @
between the top of the flights and the sleeve, where the velocity gradient is very high, because the particles were of the same order of magnitude as the clearance. However, evidence of a component perpendicular to the helix of the screw in the direction of the pressure gradient was found. Resultant Velocities. Only the components in the two principal directions have been discussed; their resultant is the velocity vector that was measured directly. A three-dimensional configuration seems the best way to show variations in magnitude and direction of the velocity vector, V, and its components, V , and vb, with depth. With maximum flow (Figure 7 , u ) the magnitude of the vector varies from a maximum value obtained at the top of the groove to zero obtained a t the bottom; in this interval its direction rotates progressively to about 90'. The plane perpendicular to the axis of rotation of the glass tube was used as reference (0'). In the intermediate flow condition (Figure 7, b), while the magnitude still varies continuously from maximum at the top to zero a t the bottom of the groove, the range of variation of direction goes from zero to about 150'. With no output (Figure 7, c) the direction of the vector remains the same a t all depths, while the magnitude and the sense change, The sense is inverted not by a rotation of the direction but by progressively changing from positive to
negative velocities. Therefore, for each condition there is a corresponding maximum rotational angle which varies continuously from 90' for maximum flow to 180' for no output. Obviously these conclusions do not hold for the zones close to the walls of the groove where. because of the reduction of the binormal component, rotation angles different from those previously mentioned are set up; the rotation is less than 90' for maximum flow and more than 180' for the no-output condition. Conclusions
The isovels, drawn according to the experimental measurements (Figure 4)) are in good agreement in all three working conditions with those obtained theoretically for tangential flow by Grant and Walker (5) and by Carley and Strub ( 3 ) . The experimental work made possible the study and representation of flow transversal to the groove, for which a theoretical and mathematical discussion was not known, although it had been suspected from logical considerations by several workers. The existence of this component, having a zero line distinct from that of tangential flow in all practical working conditions-Le., with maximum or intermediate flow-ensures a constant mixing action. Where the tangential component is zero, the layer that would be stationary is being continously changed by the circulatory VOL. 50, NO. 5
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shear motion (in the ideal conditions when the pressure a t the head of the extruder is zero) and from a backward motion due to the pressure drop between the head of the extruder and the feed port. Acknowledgment Figure 7. Perspective representation of velocity vector V and its two components V t and V, a. b. C.
Maximum flow Intermediate flow No output
C motion caused by the binormal component. Obviously the magnitude of the angle fl of the helix has more or less an important effect on the mixing action. I t is reasonable to assume that screws with two or three leads are more favored partly because of the mixing action caused by the increased binormal component, even though there is a decrease in the tangential component. The experiment cited by Rigbi (77) has a logical justification, as it was performed under the no output conditionLe., when zero lines of the components (tangential a n d binormal) are coinci-
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dent, causing a stationary layer. A further interesting experimental result appears to be the independence of the tangential and binormal flows in the groove. This allows application of the principle of summation of effects, both in the theoretical discussion and in the analysis of the experimental results obtained with a n extruder. I t also substantiates the extension of this principle (already justified by physical and rheological considerations and applied by several authors) to tangential flow. This flow can be considered in all cases as resulting from a forward viscous
INDUSTRIAL AND ENGINEERING CHEMISTRY
The authors wish to express their gratitude to Pirelli S.p.A., for permission to publish this study, and to Stefan0 Oberto and Valentino Zerbini for their valuable suggestions.
Literature Cited Boussinesq, M. S., J . des math. @res et appl. 13, 377 (1868). Carley, J. F., McKelvey, J. M., IND. END.CHEM.45,989 (1953). Carley, J. F., Strub, R. A , , Ibid., 45, 970 (1953). Eccher, S., Zbid., 43,479 (1951). Grant, D., Walker, W., “Plastics Progress,” Ph. Morgan, ed., p. 245, Iliffe and Sons, Ltd., London, 1951. Houwink, R., ti-. by G. Ricca, “Materie Plastiche,” p. 229, Hoepli, Milan, 1946. McKelvey, J. M., IND.ENG.CHEM. 45,982 (1953). Maillefer, Ch., Rev. gtn. caoutchouc 31. 563 11954). (9)1 Marietti, h., Glorn. chim. ind. afipl. 6, 567 (1924). Pigott, W. T., Trans. Am. Soc. Mech. Engrs. 73, 947 (1951). Rigbi, Z., Brit. Plastics 23, 100 (1950). Rogowsky, Z . M., Proc. Znst. Mech. Engrs. (London) 156, 56 (1947). Rowell, H. S., Finlayson, D., Engineering 114,606 (1922). Zbid., 126,249, 385 (1928). Tritton, F. S., J . Inst. Metals 26, 250 (1921). RECEIVED for review December 27, 1956 ACCEPTEDJuly 18,1957
