well the behavior of real extruders. Just as the ideal gas laws serve in the study of real gases, so do these equations serve i n extrusion. The quantitative description given in these papers of the flow behavior of plastics melts in the channels of extruder screws does not necessarily apply directly to the rear portion of a n extruder where the plastic enters the screw channel as small solid particles. Attempts to describe the motion of these solid particles have been only partially successful to date. In general, i t appears that if the solid particles are small relative to the depth of the screw threads-i.e., particle diameter 1/4 thread depth-r if for some other reason the frictional coefficient of the material against the extruder casing is small in comparison to the shear resistance of the layer of plastic particles in the screw channel, shearing does not occur in the layer and very different mechanisms may be operating. In some cases the material rides the
screw and does not progress down the screw thread channel, In other designs a material may ride the extruder cylinder walls and progress down the channel a t the extremely high rate of one thread turn per revolution of the screw. The well known trick of threading both the cylinder and screw a t the rear of an extruder ensures t h a t the solid material will be sheared and hence progress down the threaded channeIs. The papers that follow relate to conditions after the plastic has softened, and the operation is then described as “melt extrusion.’’ The combined operation of melting and extruding is called “plasticating extrusion.” In successful screw designs for plasticating extrusion, the capacity a t the rear of the screw is equal to or greater than the capacity of the melt extrusion or forward part of the screw. In these extruders the melt extrusion part of the operation controls the output and the pressure generated. Therefore the melt extrusion flow theory enables one to calculate with considerable reliability the pressure and output characteristics of any extruder.
W. L. Gore
Basic Concepts of Extrusion T h e growing interest in extrusion of plastics stimulated an investigation of the literature of this subject. The basic notions of the behavior of viscous liquids in screw extruders are presented. The historical development of the work leading up to this symposium is briefly surveyed, and the principal findings are presented. The foundation for the development presented in the succeeding papers (2, 5 ) was laid by Newton, Poiseuille, and others. As early as 1922 Rowell and Finlayson (13) had derived equations of flow for screw pumps. J. F. CARLEY
AND
R. A . STRUB’
Polychernicals Department, E. I . d u Pont de Nemours & Co., Inc., Wilmington,Del.
I
N T H E last century a great deal of fundamental work has been done on the laminar flow of viscous liquids. This m-ork has laid the foundation for the present theory of extrusion, and it is reviewed in this paper. It was recognized very early that a moving plate separated from a fixed plate by a viscous liquid will drag along with it a certain amount of fluid. Nen ton first postulated that the shear stress, 7, b e h e e n tv,o adjacent layers in relative motion is proportional to the velocity gradient or rate of shear, d s / d y . r =
pds/dy
(1)
A liquid whose flow behavior satisfies this equation is said t o be Sewtonian, and the constant of proportionality, p, is called the viscosity coefficient or simply the viscosity of the liquid. It has long been knonn that the viscosities of liquids are inversely related to temperature. I t is only relatively recently that we have found that for many liquids the quantity, p, is not constant but depends on the shear rate or shear stress. I n some liquids, the value of p is related also t o the length of time for which the liquid has been exposed to the stress. Molten plastics are the liquids usually dealt with in extrusion and many of these are non-Nevtonian. I n a surprisingly large fraction of practical cases, however, it has been found that the flow relations established for I
Present address, % Sulzer Bros., Winterthur, Svvitzeiland.
970
Semtonian fluids can still be used providing a suitable “apparent” viscosity is introduced. .4n extruder consists of a threaded shaft or screw rotating in a fixed cylinder or barrel, plus a feed port and a die. The screxchannel is constantly changing its position relative to the barrel, and the advance of the liquid through the barrel of the extruder follows a helical path which is a mirror image of the helix on the screw. This situation is rather difficult t o visualize. The end results mould be the same if the screw were held stationary and the barrel rotated in the direction opposite to the actual rotation of the screm, since the relative velocities of screw and barrel are unaltered. But in this case, the liquid moves along the helical path defined by the screw channel. Because the channel walls are attached t o the screw, it is far easier t o visualize and discuss the situation in which the barrel rotates. For this reason, the ?crew is taken as the frame of reference throughout this symposium. The upper picture in the first figure represents a portion of a n extruder. The speed of rotation of the barrel, U,is resolved into two components, T and V,perpendicular to the direction of the screw thread and parallel to it, respectively. The flow pattern within the channel cross section is a combination of four types of flow. The first is the drag flow, QD, in the direction of the axis of the channel created by the velocity com-
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 5
Extrusion
f TRANSVERSE
FLOW
V
maximum pressure is developed. What this means is t h a t the drag flow is exactly offset by the pressure flow and leakage. The top sketch of Figure 1 shows the path of molecules of the melt in the helix. Except at zero discharge it is a helix within a helix. If the helix angle is small, then the transverse component will be smaller than the forward component, so the rate of advance of the inner helix will actually be much greater than is schematically shown in the sketch. For example, if the pitch of t h e screw equals the diameter, the average molecule will advance about two turns of the screw while completing one turn of the inner helix. The sketches in Figure 1 also define the dimensions, D, h, b, S, e, and y. The two smaller sketches show lines of constant velocity (isovels) for drag and transverse flow, projected on a plane perpendicular to the axis of the helix. Figure 2 shows similar isovels for the component directed along the helix. The left-hand sketch shows isovels for zero discharge; the numbers are fractions of the maximum speed, V . HISTORICAL DEVELOPMENT
SECTION 0-A
Figure 1. Sectional Views of an Extruder, Showing Dimensions and VeIocities
ponent, V , of the moving plate. The second type is the transverse flow caused by the component, T. The opposing pressure flow, &, directed backward along the channel is the third kind. This pressure flow or back flow is caused by restricting the free discharge of the liquid t h a t is dragged forward. I n extruders we do this by attaching a die t o the end of the barrel. A pressure is built up at the front end which causes some of the melt t o pass through the die and some of i t t o flow back along the channel. The fourth type of flow is the leakage flow, QL, between the barrel and the tops of the screw flights directed backward along the axis of the screw. This flow is also caused by the buildu p of pressure from the rear t o the front of the screw, from one turn of the thread t o the next. When there is no restriction, the extruder is said to be at “free discharge” or “open discharge,” and the pressure flow and leakage a r i both Zero. If the end of the machine is cbsed Figure 2. off completely, there is no discharge, and the
The differential equations governing all four types of flow were first established by Navier in 1822 (8). Long before the invention of screw pumps, the problem of pressure drop and flow in tubes and pipes got much attention. The pressure flow in an extruder is a special case of pipe flow in which the pipe has a rectangular or semielliptical cross section. The differential equation of flow as presented by Navier is given (Equation 2 ) .
MAXIMUM FLOW
NO FLOW
Enlarged Cross-Section of Screw Channel Showing Lines of Constant Velocity
NOMENCLATURE FOR EXTRUSION SYMPOSIUM
b C D
-
e E h k
L n N Ap
AP
Q QD
QL QP ~
thread width, measured in direction of screw axis heat capacity of polymer mean diamete’r of screw = root diameter plus thread deDth: in a shallow screw maior diameter approximates -_ D hoiely enough = land width, measured in direction of screw axis = (1) eccentricity of screw in barrel; (2) pumping efficiency of extruder = thread depth = dimensional constant of die equation; briefly, die constant = axial length of flighted section of screw = number of flights in parallel in a multiflight screw = rotational speed of screw (revolutions per unit time) = difference in pressure from one side of flight t o other = rise in pressure along screw. If static ressures at feed port and discharge point are equal, A$ equals drop in pressure across die = volumetric rate of discharge of screw = volumetric drag flow along channel = volumetric leakage flow across flights = volumetric pressure flow along channel = = =
~~
~
~~
May 1953
t
U V 2,
W X
P ‘p
+
lead of screw. t = n ( b e ) . I n a single-flight screw t = pitch, also = peripheral velocity of barrel, relative to screw (or vice versa). U = d D N . = component of U directed along helix = local velocity of a particle in flight a t point (2,y), positive when directed in positive z direction = width of channel measured at right angles t o axig of helix. w = b COS ‘p = direction perpendicular t o depth and axis of channel; x varies from zero t o w = direction of depth of channel; y varies from zero at screw surface to h at barrel surface = direction along axis of helix, increasing dieward = total power required to turn screw = clearance (radial) between top of flights and barrel = a variable distance coordinate, measured along screw axis and increasing dieward = viscosity of melt = helix angle = arctan t / r D =
~
INDUSTRIAL AND ENGINEERING CHEMISTRY
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where z is distance measured along the helical axis of the channel, and v is the velocity of any point in the channel, in the steady state. I n 1868 Boussinesq ( 1 ) solved this differential equation, introducing the correct boundary conditions for the pressure flow in pipes of elliptical and rectangular cross section. His mathematical solution confirmed the experiments niade in 1842 by Poiseuille on viscous flow in capillary tubes (8). Integration of the differential equation gives the local velocity in the thread as a func-
This equation resembles Equation 2 save that the pressure gradient along the channel is zero. However, the boundary conditions are different, since the relative movement of the surfaces of barrel and screw is the prime mover, rather than hydrostatic head. Here if the local velocity is integrated over the whole cross section, the volumetric drag flow, QD, becomes =
QD
aDhW cosa 9 FD
m
FD =
-$E[ 1 cash (gap) - 1 ga sinh (gap)
(7)
...
0 = 1,3,5
where F D = a shape factor for drag flow, plotted in Figure 4. What of the transverse component of velocity, so far neglected? Since this component is directed across the channel, it contributes nothing to the output. On the other hand, when heat is t o be transferred to or from the polymer, this transverse component provides forced convection in a liquid so viscous that natural convection is negligible. Grant and Walker (4) have demonstrated t h a t the transverse component is also important t o the successful mixing of pigments and fillers v i t h the polymer. os
1
T---
04
0.1
0.2
0.3
0.4
OS 0.8
2
I h
3
4
5
6
1
1
RECTAHQULAR PROCILC
/
-
810
b.cos7 Figure 3.
Shape Factor for Pressure Flow
tion of the two distance coordinates perpendicular t o the axis of the channel, If this velocity distribution is itself integrated over the cross section of the channel, the volumetric pressure flow is obtained. Equation 3 for QP was derived by the present authors. It is equivalent to Boussinesq’s solution but much simpler in form and easier to work with. QP
bh3 COS p FP
p
= ___
d_P dz
YE[+ ($)I
(3)
m
_ --
FP
1 12
tanh
B = 1,3$
(4)
...
where p = h / ( b cos p ) and F p is a shape factor for pressure flon-, dependent on B. Shape factor F p is plotted for rectangular and semicircular channels in Figure 3. It is interesting to note t h a t the basic differential equation of viscous flow has the same form as the one which describes the torsion of a prismatic bar and that this problem had already been solved by Saint-Venant in 1855. The case of visrous flow through semielliptical thread profiles also has analogies in laminar heat transfer, torsion, and the deformation of thin membranes ( 9 ) . During the years 1922 and 1928, Rowell and Finlayson ( 1 3 ) applied the general theory of lubrication developed earlier by Reynolds (IO), who recognized the hydrodynamic nature of lubrication. They gave a solution of the homogeneous differential equation describing a pure drag flow in an extruder:
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h b.cos’t
Figure 4.
Shape Factor for Drag Flow
The leakage past the flights as previously noted, is a form of pressure flow, and it therefore must behave according to the differential equation for pressure flow. I n this case the geometry is a little different. Here the liquid is flowing through a very thin ring between land and barrel. The two-dimensional equation reduces to a one-dimensional equation since the annulus may be thought of as a long narrow slit without ends. These boundary conditions were applied to the differential equation by Rowel1 and Finlayson who obtained the equation for the leakage flow: QL =
TDEPAP ~
12 e p
The factor, E, is a correction for possible eccentricity of the screw flights in the barrel, and it varies from 1 for a perfectly centered sere-- to 2.5 for a screw that is rubbing on the barrel. Because there are so many flights on the ordinary screw, it seems likely that if even a few of them were off center, there would be galling of the screw and barrel. A value of E near 1, say 1.2, is reasonable for general use. If the density of the melt is unaffected by pressure and if no material is created or destroyed in the extruder, the output of the extruder must be given by
Q
= QD
- QP - Q L
INDUSTRIAL AND ENGINEERING CHEMISTRY
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Vol. 45, No. 5
Extrusion
Y
This is called the equation of continuity for extruders. Imagine a n extruder with a die whose characteristic can be changed over a wide range-for example, a valve could be used in place of a die. Then for every valve setting there would be a given output and a corresponding pressure. The graph of pressure versus throughput is called the pressure-discharge characteristic of the screw. If the liquid is Newtonian and is a t the same temperature in all parts of the extruder, this characteristic is a straight line. This was first pointed out by Rowell and Finlayson ( I S ) , and it is in agreement with what has gone before. When there is no pressure (the condition of free discharge), the discharge is equal t o the drag flow. Thus, &D is the intercept of this line for the ordinate, pressure = 0. As the pressure mounts, back flow and leakage climb in direct proportion, reducing the discharge linearly. Rowell and Finlayson discussed extrusion tests made with several oils and water solutions, and their results agreed closely with the theoretical predictions of pressure and output. They also proposed approximate expressions for the power needed t o turn the screw and for the efficiency. Their work appears to have gone unnoticed until 1951, when it was cited by Pigott ( 7 ) ,who reported on his experiments with oils and rubber. Pigott also attempted to take into account the effect of shear rate on the viscosity, but there were several serious mathematical errors in his work. I n 1947, Rogowsky (la), apparently unaware of the work of Rowell and Finlayson, published a similar derivation of the extruder flow equations. His paper, too, contains a serious error, which is responsible for the odd-loolfing velocity distribution he gave. Rogowsky also realized that at zero discharge there should be a plane of zero velocity in the screw, and later under the name Rigbi ( I I ) , he described an observation in which this deduction was confirmed. H e pointed out the possible serious consequences of extruding heat-sensitive vinyls a t back flows so high that the layer of zero velocity existed. He probably forgot that the transverse flow does much to keep polymer molecules moving through the region of low velocity. Grant and Walker (4) gave some velocity distributions in screw threads and discussed screw efficiency. The volumetric efficiency of the screw may be defined a s the volume delivered per turn of the screw divided b y the volume of the screw thread per turn. Grant and Walker stated t h a t the maximum efficiency is but failed to realize t h a t the efficiency is a function of the channel shape. The efficiency of the deep channel they studied (h = 0.8 b) is about 27%. But in shallow channels (h = 0.1 b ) the efficiency is about 4570, and for constant viscosity in the thread the limiting value, in very shallow channels, it is 50%.
Even higher values can be reached if there is a positive viscosity gradient in the y direction, for example, when the material at the root of the thread is hotter than t h a t near the barrel. Eirich (3)considered screw extruders as machines for developing high pressures, and derived an expression for the maximum pressure attainable. I n a recent paper, Strub (14)discussed the influence on output of varying viscosity across the thread depth. H e also presented laws of similitude, as well as a formula for the power absorbed, slightly different in form but otherwise identical with the one in reference (6). The Nomenclature section of this paper lists the symbols t h a t are used throughout this symposium. X o units are given here; rather, they will be introduced when needed. Except where t h e contrary is specifically stated, any consistent system of units may be applied to the equations presented in this symposium, with one exception. The exception is that angular displacement of the screw is always measured in revolutions rather than radians, the customary mathematical unit. There are a few symbols t h a t do not appear on this list which are used in one place onlv. Here and there some symbols are used to represent quantities which are discussed in one paper only, and they are defined on the spot in each case. LITERATURE CITED
(1) Boussinesq, M. J., Math.
~ U T appl., ~ S Second Series, 13, 377 (1868). (2) Carley, J. F., ihllouk, R. S., and McKelvey, J. M., IND. ENG. CHEM.,45, 974 (1953). (3) Eirich, F. R., I n s t . Mech. Engrs. (London),Proc. 156, 62 (1947). (4) Grant, D., and Walker, W., paper in “Plastics Progress” (ed., P. Morgan), London, Iliffe and Sons, Ltd., 1951. ( 5 ) Mallouk, R. S., and McKelvey, J. M., op. c&, 987. (6) Navier, C. L. M. H., Mem. Acad. Sci. (Paris), 6 (1822). (7) Pigott, W. T., Trans. A S M E 73, 947 (1951). (8) Poiseuille, J. L. M., Compt. Rend., 15, 1167 (1942). (9) Purday, H. F. P., “An Introduction to the Mechanics of Viscous Flow,” New York, Dover Publications, Inc., 1949;’ published in England under the title “Streamline Flow,” London, Constable and Company, 1950. (10) Reynolds, O., Trans. Roy. SOC.,A , 177, 157 (1886). (11) Rigbi, Z., Brit. Plastics, 23, 100 (1950). (12) Rogowsky, 2.. Engineering, 162, 358 (1946); republished in Inst. Mech. Engrs. (London),PTOC.156, 56 (1947). (13) Rowell, H. S., and Finlayson, D., Engineering, 114, 606 (1922); 126, 249, 385 (1928). (14) Strub, R. A., Proceedings of the Second Midwestern Conference on Fluid Mechanics, Ohio State University, March 17-19, 1952.
RECEIYED for review October 21, 1952.
ACCEPTEDMarch 6, 1953
Extrusion Experimental Equipment
May 1953
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