I
M. 1. JACKSON',
F. J. LAVACOT, and H. R. RICHARDS
U. S. Naval Ordnance Testing Station, China Lake, Calif.
Screw Extrusion Theory with Application to Double-Base Propellant Data on solids conveying in screw extruders aid in designing presses for solid and semisolid flow and in scale-up of propellant extruders
Tm
SCREW PRINCIPLE is useful not only for simple conveying where zero net pressure is developed, but also for pressures of many thousands of pounds. I t is applicable for true liquids of low viscosity, highly viscous plastic materials, and granular solids, either coarse or finely divided. Continuous processing is possible, with good control and product uniformity. The theory of the screw press has been most completely developed for materials 70, 74). that behave as true liquids (4,9, However, many materials may behave as solids or semisolids; in the work reported here, the theory for such non-Newtonian materials is discussed.
For either case the pressure increase in a material moving as a solid with slip is exponential with length. Obviously, the drag at the plate must exceed that at the channel walls-Le., C, > k'C,or the material will not move. The channel shape, characterized by k', and the coefficients of friction between the material and the surfaces determine rate of increase of pressure with length. Hence, these items are important in describing the flow of materials through screw-type processing equipment where slip is a factor. Where the flow is viscous, the pressure increases linearly with length of flow channel. Hence, the presence of slip may cause a radical departure from typically viscous flow.
Solid Flow The assumption that no movement of material occurs at the confining wall appears valid for materials having a viscosity lower than the frictional forces a t the wall. However, for highly viscous materials or granular solids, slip at the wall commonly occurs. Consider the material confined between a moving plate and a stationary channel having side walls. As for liquids, it is assumed that the pressure is uniform across any section. For a differential section of material of thickness dx and with slip relative to the channel only, the frictional force a t the channel walls is balanced by the force arising from the pressure drop across the section, or C a p (w
+ 2 h ) dx = S d j = h w d j
Rearranging, and imposing limits over the channel length,
or In p&,
= k'C,x/h
(1)
For slip also at the plate a similar analysis gives l n j a / j > = (C, - h-' C.)
-f
(2)
1 Present address, Department of Chrmical Engineering, University of Idaho, Moscow, Idaho.
Screw Theory for Solid Flow The geometry of a helical screw channel complicates the equations for flow rate and pressure increase but can be accounted for by introducing an additional variable. One characteristic of flow through a screw is the angle between the thread and a plane perpendicular to the screw axis. This angle and the dimensions of the channel define movement for viscous flow. However, for solid flow, slip occurs a t the wall and as the material partially rotates with the screw, it is displaced in the direction of flow. A point in the material describes a helical path opposite in direction to that of the screw helix. The angle of the helical path of the material is the additional variable which must be introduced to describe the flow. Maileffer (8)chose an angle which had an elusive physical significance. The material angle is better defined as that between the screw helix and the material helix, represented by W. Several assumptions are necessary: The coefficients of friction do not change with pressure, any section of material moves as a solid block (no velocity gradient), and pressure is transmitted as in a fluid. Throughput for Solid Flow. A section of material has a speed and direction as indicated by V,, in Figure 1, a. The material velocity has a component, V,. in the direction of the press axis which
represents the movement of the material through the press. Velocity (relative to the cylinder) may be found as the vector sum of the velocity of the material relative to the screw, V, (ih the direction of the screw helix), and the velocity of the screw relative to the cylinder, V,. V, arises because of slip of material relative to the screw and V , arises from rotation of the screw. The output of the press is given by Q = V,S
(3)
which is related to the material velocity by the right triangle in Figurc- 1, a V, = V, sin ( w - 4) = V, (sin w cos - cos w sin +)
+
(4)
Applying the law of sines to the obtuse triangle
+
V,/sin
= V,/sin (180 - w ) = V,/sm w
and V,
=
( V , sin +)/(sin
w)
(5)
A-oting that V, = irDN and eliminating V , and V , from Equations 3, 4, and 5 Q =
r
= r
DNS sin2+ (cot + - cot w ) DNS sin + cos + (1 - tan + cot
w)
(6)
Equation 6 may be written in terms of the flight volume, V,l, the product of the cross-sectional area of the screw channel perpendicular to the screw axis and the pitch or advance of the screw for one revolution, or V,, = S ( r D tan +)
(7)
This result may also be obtained from Equation 6, because for maximum production (no rotation of material in the 4, cot w = screw direction) w = 90' cot (90' 4 ) = -tan 4. The extrusion equation for solid flow thus becomes
+
+
Q = VfiNcos2+(1 - tan+cotw) (8)
and press output is related to the screw displacement through the material and screw helix angles. For w = @, the minimum value w can assume, the press output is zero and the material rotates with the screw without any net forward movement. Equation 8 may be exVOL. 50, NO. IO
*
OCTOBBR 1958
1569
Figure 1:
Relationship of Velocities and forces to angles for solid flow
pressed in terms of cy for shallow screws for which V f l = 2cy/cos2 4. Pressure Development for Solid Flow. The relationship among pressure? screw length, and the material angle is determined from a force balance. The frictional force exerted on the material by the cylinder, F,, is balanced by the forces from the pressure gradient. F,, the frictional force arising from pressure on the screw surfaces, F,, and the additional frictional force, F,. The last force occurs on the pushing side of the screw channel and is induced by the drag at the cylinder. The forces acting on a differential section of material are (Figure 1, 6): dF, = (;b w dx) C,
dFt = dF, dF,
COS
= ;b
(90 -
p
w
(w dx
dFp
=
(9) C, = sin w C, C, dx ( 1 0 ) w)
+ 2 h dx) C, dp h
u,
(11) (12)
Equating forces in the channel direction, dF,
COS
w =
dF,
+ dF1 + dF,
substituting Equations 9 through 12. and noting that dx = dL/sin 4, the equation for pressure development is obtained d In p / d L = C,
COS w
- C, (w
+
2 h ) / ( w ) -C,C, sin w h sin 4
exponentially with screw length. A plot of log ;bd us. length of pressure development in the screw will be a straight line if ffis constant and the coefficients of friction and the material angle are independent of pressure and screw speed. Significance of Equations. Equation 14 indicates that the maximum pressure for a given screw length occurs for the minimum value which the material angle can assume; from Figure 1, a, this is for w = +. Thus, maximum pressure development increases as the screw helix angle decreases and for large pressures small screw helix angles are indicated. Back pressure imposed by a die may become so great that flow ceases (Equation 15). The factors which determine the material angle can be visualized by rearranging Equation 13 to cos w - C, sin w = h sin 4 ( d l n p )
-4[2]
[(T)
+
(15)
As the back pressure is reduced, the right side of Equation 15 becomes smaller. Because normally C, is small, the net effect is an increase in the material angle and throughput increases. For no net pressure developed, as in a screw conveyor, ( d In p)/(dL) = 0: and cos w = k'C,/C,
+ C, sin w
k ' C,/C,
(13)
(16)
For a length of screw channel, L , considering the pressure at the feed end, pf, to be finite and substituting the shape factor, k', for (w 2 h ) / ( w ) , integration gives the pressure a t the discharge end, p d ,
This is the maximum value w can attain and is determined by channel shape and coefficients of friction. The output of the press is determined accordingly by Equation 8 and represents the true maximum. The material angle cannot attain the value of G' = (90' 4 ) . According to Equation I6 the material angle approaches 90' as a limit as C, approaches zero. This is also observed by inspection of Figure 1, b, where w cannot exceed 90°-that
+
C, cos w - h' C, - CcC, sin w] h sin 4
+
(*4)
[ E -
For a press of fixed design and extrusion conditions, the pressure increases
1570
INDUSTRIAL AND ENGINEERING CHEMISTRY
is, the pushing force of the channel wall is perpendicular to the wall and colinear with F, The factors determining the material angle, then, are the coefficients of friction, and to a variable degree, the back pressure imposed by the die. Under some conditions, the latter has the least effect. The effect of channel shape may not be great and k' can be greater or less than unity. For very shallow screws, k' = 1 ; as the depth is increased the root surface of the screw decreases but the side wall area increases. Depending on the pitch of the screw thread, the area of the screw surfaces may be greater or less than the area at the cylinder. Values of k' for the deep screws employed in this investigation were close to unity (Table 11). Maileffer (8) developed similar equations for solid flow, but incorrectly omitted the frictional force, F,, in the analysis. Darnell and Mol (5) give a rigorous derivation of the pressure development equation using torques instead of forces. The resulting equations are so complex that design charts are given to facilitate calculations. For present purposes, Equation 15, though based on an approximation, is preferred because it indicates simply factors that influence the material angle. The use of k' accounts to some extent for the fact that the screw channel is helical rather than straight. Darnell also gives an equation for throughput. similar to Equation 8. However, Equation 8 is preferred because the movement of the solid mass through the press is expressed as a single function of the material angle ds defined. Press Design for Solid Flow. Equation 8 indicates that the material angle should be large for high throughputs, which from Equation 16 requires the coefficient of friction at the screw to be small but that at the barrel as large as practical. The frictional drag at the screw surface is determined by the type of material being processed and the material of construction employed. Surface coatings to reduce the coefficient of friction at the screw can be used to increase the material angle and throughput. Output can be improved by increasing the drag a t the surface of the cylinder. Longitudinal grooves or flutes in the cylinder walls are effective because material flow in the direction of screw rotation is reduced and easy movement along the axis of the cylinder toward the discharge end is still possible. The coefficient of friction for a fluted barrel is a composite of frictional drag and resistance of the flutes. For the latter. shear resistance of the material may contribute to the apparent coefficient of friction at the cylinder. Methods for determining the coefficieiiL of friction of granular materials under sliding conditions have been reported
SCR6W HXTRUSION THEORY (73, 75). Coefficients were influenced significantly by the presence of a lubricant, such as a plasticizer. Coefficients of friction of plastic materials are reported to vary significantly with temperature; this effect has been observed in extrusion presses operating with slip. An approximate value for the screw helix angle to give maximum output may be obtained by assuming the material angle to be determined by Equation 16. Equation 18 is differentiated and (dQ/dd) is equated to zero: +Opt.
Z w/2
1/2 c0s-l ( k ' C,/Co)
(17)
Thus, maximum output will be approached when the material helix is the mirror image of the screw helix. The optimum thread depth is not readily obtained. Output increases with increasing depth (approximately S = nDh) (Equation 6), but this must be modified by the fact that the material angle decreases with increasing thread depth (Equation 15). If, the ratio of the coefficients of friction were known, a trial solution would indicate an optimum value. Deep threads are probably indicated, because back flow does not occur. Smaller values of (Ca/Co) permit deeper optimum depths. Strength considerations and, in some cases, cooling or heating passages impose limits as to thread depths. Leakage flow through the clearance will not occur for spacings less than the particle size if granular materials are extruded, and probably clearances several times this size can be used. This simplifies construction and offers one means of reducing or eliminating rubbing of the screw on the cylinder wall. Rubbing is encountered in single-flighted screws because of the unbalanced force a t the delivery end of the screw where the thread terminates. Double-flighted screws do not always eliminate rubbing, because feeding of parallel channels is sometimes uneven. Screw lengths can be relatively short even for the development of high pressures, if fluted cylinders are employed. Few design problems will be encountered with respect to screw length. Semisolid
Because of viscous deformation the slip velocities relative to screw and cylinder will no longer be the same. The velocity gradient for viscous flow is modified because of slip. The situation becomes too complex to be treated mathematically. Viscous flow with slippage at the wall has been described in terms of a lubricated layer adjacent to the wall (Z), but this mechanism is of no value for semisolid flow. Semisolid flow involves viscous and solid flow and these represent limiting types of behavior (Table I). I n some cases, behavior depends more on slip than on viscous characteristics. True viscous behavior would be approached for comparatively low viscosities; conversely, flow would more nearly correspond to solid flow for large viscosities, where shear stresses in the material exceed frictional stresses a t the walls. With slip the exponential increase of pressure with length predominates; the linear gradient arising from viscous flow has little significance. Hence, when slip occurs, screw lengths can be short. For solid flow the optimum screw helix angle was indicated to be approximately one half the material angle. For viscous flow it is 4 = 30'. A first approach for combined flow would be for the optimum screw angle to equal one half the material angle. Experimentation may show that some departure for this toward = 30' would improve performance. For viscous flow, flight depth is of critical importance because of back flow from the die to the rear of the screw caused by the pressure gradient. This becomes of lesser importance as the viscosity increases. Designs intermediate between the optimum depth indicated for viscous flow and for solid flow
+
Table 1. Characteristics Volumetric discharge rate Pressure development Maximum screw efficiency
Flow
The forces at the wall may be so great that slip occurs but, a t the same time, the material may undergo viscous shear. The forces acting a t the screw and cylinder surfaces impose a shearing stress in the confined material. If the yield point is exceeded, viscous flow will occur. The combination of viscous flow with slip might suitably be termed semisolid flow, as it exhibits more the characteristics of solid flow.
Helix angle for maximum throughput Flight depth for maximum throughput Clearance Screw length
would serve as a starting point as judged by the magnitude of the viscosity. Similar considerations hold for the selection of clearance between the cylinder and screw thread. The coefficient of friction of a semisolid material, or a viscous material which undergoes slip, may depart significantly from that for the same material in a rigid state. When the material undergoes deformation, it can conform to the microscopic contour of the confining surface. The effect of flutes in the cylinder becomes less as the shear resistance of the material decreases. For no slip at the walls, fluting offers no advantage. The coefficient of friction of a semisolid or highly viscous material cannot be determined experimentally by the usual methods. Keeping the material under confinement at press conditions of elevated temperature and pressure is difficult. Actual performance in a screw press may be the only feasible way to estimate C,/C,, particularly as C, is a composite or effective coefficient of friction, Toor's method (75) might be adapted for use with semisolid materials. Extrusion of Double-Base Propellant The extrusion of double-base propellant in a screw press illustrates semisolid flow. Slip occurs at the walls of the press and die channels, and viscous deformation is observed. T h e material has a rather high viscosity and shows the presence of a yield point below which deformation does not occur. Data for the performance of a screw press operating on a material of this type have not been reported previously. Double-base propellant provides a case of extrusion where viscosity is large and
Summary of Design Characteristics for a Screw Extruder Viscous Flow (3 y) (APP/r) Pressure increase is linear with screw length 50% for all drag flow; decreases with back and leakage flow = 30°
Q = aN
- +
+
Semisolid Flow Nomathematicalexpression available Press. increase more nearly exponential with length Maximum could range between 50 c$
>
Solid Flow Q = Vf2 N cos2 q, (1 - tan cotw) Pressure increase is exponential with screw length
