I
W. D. MOHR and R. S. MALLOUK‘ Polychemicals Department, E. I. du Pont de Nemours & Co., Inc., Wilmington, Del.
Flow, Power Requirement, and Pressure Distribution of Fluid in a Screw Extruder Simplified equations make the extrusion theory more usable and understandable by a quantitative description of transverse flow
THE phenomena involved in the extrusion of viscous liquids by screw extruders were described and quantitative relationships outlined in a Symposium on the Theory of Plastics Extrusion (7-8). Further examination of this extrusion theory has indicated that the simplified derivations could be made more rigorous by additionally considering the flow and pressure distributions in the transverse plane in the screw channel. Furthermore, the effect of clearance between land and barrel surface makes itself felt primarily as a diminution of forward flow in the channel, as well as “leakage” flow back over the lands. This discussion presents the simplified extruder flow equations, taking into account this effect, a corrected derivation of the power equations, and the power equations compatible with the flow equations.
moving barrel surface and the flow in the opposite direction under the influence of a pressure gradient. The motion of the fluid in the transverse plane is termed “transverse flow.” The flow of fluid across the top of the land from one channel to another is termed “leakage flow” ; this flow results
\
from the pressure difference on the two sides of the land. The discharge from an extruder is the net volumetric rate of flow across any surface intersecting the barrel of the extruder. I n preceding studies ( Z ) , a plane perpendicular to the channel was selected for the calculation of drag flow
X
Flow and Pressure Distribution The mechanism of viscous flow through the screw channel can be visualized by “unwrapping” the channel and lands from the screw and laying this channel flat (Figure 1). The barrel surface is then flattened and caused to move across the top of the channel with a velocity U at an angle 6 to the direction of the channel. The liquid in the channel is assumed to wet all surfaces and to be moved by the shear stresses developed by the relative movement of the barrel and channel. For convenience, the velocity of the barrel relative to the channel is resolved into two components: One component, V, is directed along the direction of the channel; the other component, T , is directed in the plane transverse to the direction of the channel. Flow in the direction of the channel results from the flow in the direction of V induced by the Present address, FilmDepartment, E. I. du Pont de Nemours & Go., Inc., Wilmington, Del.
Fig. la X’
Region B
Fig. Ib
Section K
46
/
-x’
Region A
Figure 1. The geometry of the “unwrapped” screw channel and barrel surface is presented in this diagram VOL. 51, NO. 6
JUNE 1959
765
V
c Flow
Figure 2. T h e addition of the velocity profiles for d r a g flow and for
P r e s s u r e Flow /
/
/
/
/
/
v
/
/
pressure flow gives the resultant velocity distribution down the channel
The parameter, h, is defined as the radiadistance from screw root to barrel sur1 face; this definition differs from that of the previous symposium (5) and is considered more fundamental. The first term on the right side of Equation 3 is the drag flow velocity, and the second term is the pressure flow velocity. A typical example of these flow profiles is shown in Figure 2. Equation 3 may be simplified by defining
and a =
/
&; (g)
(5)
Equations 3, 4, and 5 may be combined to give
R e s u l t a n t Flow
u =
V[(1
- 3a)H + 3aH2]
(6)
At any point in the channel, h, V , p, and dP/dz have finite values; a is therefore a determinable value and characterizes the velocity profile in the channel in the z direction as a function of the fractional distance from the root of the screw to the surface. The value of a is 0 for the case of maximum throughput with no pressure rise (dP;’dz = 0); a is 1.0 for the case of no throughput with maximum pressure rise, and a assumes intermediate values for intermediate conditions. The flow profile and pressure distribution in the transverse plane have not previously been considered. Examination of the flow in the transverse plane (Figure 3) indicates that there is a drag flow under the influence of velocity, T , but as there is only small net flow across plane A - A , a pressure gradient must exist to induce a pressure flow in the opposite direction. The velocity profile in the transverse plane may be written by analogy with the expression for flow doivn the channel
I and pressure flow. Only flow in the direction of the channel was considered and transverse flow was neglected. Finally, a leakage flow or flow back over the lands was calculated independently and subtracted from this to obtain the net flow.
Q
=
Qn -
QP
-
QL
(1)
In this study the discharge from an extruder is considered to be the net volumetric rate of flow across a plane perpendicular to the axis of the screw. From Figure l , b , this flow is seen to be the algebraic sum of the flow in the channel (region A ) plus the flow betwecn the land and the barrel (region B ) .
Q
= QA
+
QB
(2)
Flow in the direction of the output end of the screw is considered positive; Q A is therefore inherently positive, and Q B (the “leakage” flow) is inherently negative. Consideration of the transverse flow and the pressure gradients at
all points in the s)-stem is essential for such a calculation. The exact calculation of the flows in regions A and B for all conditions of screw geometry and liquid physical properties and initial conditions is exceedingly complicated. T o permit a simplified analysis of the problem, several assumptions have been made. The equipment is in steady operation. The liquid is incompressible. The viscosity is the same at all points in the system. A paraphrase of this assumption is that the operation is isothermal and the liquid is Newtonian. The width of the channel is large compared with the depth; edge effects in the fluid at the land are negligible. End effects, entrance and exit, are neglected. The velocity down the channel as a function of the distance from thread to barrel surface may be expressed (2) as
This equation is simplified by the definition of Equation 4, H = y / h , and by defining
Tra n s v e r s e Flow
Equations 4, 7, and 8 may be combined to give s = T [ ( 1- 3c)H
Transverse Velocity
I3 Figure 3, shown
766
Profile A c r o s s A-A
T h e transverse flow is circulatory with a velocity profile as
INDUSTRIAL AND ENGINEERING CHEMISTRY
+ 3cH21
(9)
This expression for the velocity profile in the transverse plane was used by hlohr, Saxton, and Jepson ( 9 ) in their analysis of mixing in screw extruders. The use of this expression permitted calculation of the mixing effectiveness of screw extruders which was borne out experimentally. The velocity in the direction of the axis of the screw, f, at any depth in the
SCREW E X T R U S I O N channel may be computed by adding together the components of velocities u and s in the direction of the axis of the screw. f = u sin 4 - s cos 4 = V [(I - 3 a ) H T [ ( 1 - 3c)H
+
3aH2] sin 4 -
+ 3cHq c o s 4
(IO)
The geometry of the system relates V and T to the dimensions and speed of the screw. V = U COS 9 = T D N
d,
(11)
T = U sin 4 = T D N sin d,
(12)
COS
u
/ I\
When Equations IO, 11, and 12 are combined, f = T D N sin 4 cos 4 X 3 X ( c - a ) ( H - P) ( 1 3 )
\
\
This expression states that the forward velocity a t a point in the channel is linearly dependent on both c and a; in addition, the forward velocity distribution with depth is parabolic in nature. T o determine the volumetric flow rate forward in the screw channel, the product of the velocity and cross-sectional area is integrated from the root of the screw to the barrel surface. Figure 4. The channel geometry i s shown with points marked for discussion of pressure gradients
- m D N h ( t / n - e ) cos2 d, 2
dP I [ c - a1
If we make the simplifying assumption that c = 1.0 for no net flow in the transverse direction, Equation 14 becomes QA
=
nirDNh(t/n - e ) X cosz 4 (1
z QD
-
- a) -
QP
dF'
t=
(14)
(15)
This expression is identical with that derived by Carley, Mallouk, and McKelvey (2) (same simplifying assumptions), who considered only flow in the direction of the channel. When the velocity components in the direction of the channel and in the transverse plane are properly summed to give the volumetric flow forward across a plane perpendicular to the axis of the screw, a more useful picture of flow in the channel is obtained. The pressure gradient that exists in the transverse plane has not been previously considered. This pressure gradient is such that the pressure flow induced virtually balances the drag flow induced by the movement of the barrel surface in the x direction. Figure 4 illustrates the unwrapped channel with several points marked on the channel to assist the discussion. The pressure gradient in the transverse plane in the channel may be computed from Equation 8.
The pressure gradient in the transverse plane is virtually independent of the back flow in the channel. (A very slight effect of back flow on c willibe shown.) T h e pressure gradient down the channel may be computed from Equation
5.
-
0
c
E n '
E
0 u
a" 3a 0
3
12
6 0.007 in. N 5 0 RPM h OO.IOOin. e 00.25 in. I
X u =
The pressure rise in the x direction over the land may be computed by noting that the pressure rise from A to B plus the pressure rise from B to C (both taken in the x direction) must equal the pressure rise in the z direction from A to C along one n'th of a turn of the helical channel.
D O 2 in. 0 0 30° n* I b 3.378 In. 580 mVmin.
--
I
I
/
/
/
1
L
Eq, 2 8 Eq, 2 7
ii I - ; 0 Y O
s
/ O V 0
1
0 ;5
I,O
Fraction Maximum Possible Pressure Ordient
(7) dz max
Figure 5. A comparison of expressions. for leakage flow is given for a representative screw geometry and o$erating condition VOL. 51,
NO. 6
0
JUNE 1959
767
dP r D cos dz n
+
(18)
Equations 16, 17, and 18 are rearranged to yield -6 p r D N eh2
+ b sin + X c ] [Yg
(19)
We are concerned with flow between the land and the barrel in the direction parallel to the axis of the screw. This leakage flow is pictured as flow through a slit under the influence of a pressure gradient. The volumetric rate of flow through a slit under the conditions assumed is directly proportional to the width of the slit, the third power of the clearance, and the pressure gradient perpendicular to the slit width, and inversely proportional to the viscosity of the liquid. The width of the slit in this case is the distance across the land(s) in the circumferential direction, ne/tan
0), a finite flow over the lands is predicted. Equation 23 contains the term p/pL, the ratio of the viscosity of the liquid in the channel to the viscosity over the land. This ratio would be unity if our assumption of constant viscosity throughout the liquid were obeyed. In the extrusion of non-ATewtonian liquid such as thermoplastic resins, however, the viscosity of the liquid falls off markedly with increasing shear rate. Because the shear rate in the clearance over the land may be ten times the maximum shear rate in the channel, may be five times as large as p L . I t is therefore useful to maintain the ratio p / p L in the expression for leakage flow, so that the effect on the leakaye flow of the reduced viscosity in the clearance may be clearly seen. Equations 2, 14, and 22 may be combined to give the total flow.
Q
=
QA
+
QB
=
nrrDNh(t/n - e ) cos2 $ (c - a ) 2
-
4.
the first terms of equations 26 and 27, which give the flow in the channel. The most prominent effect of the fact that the land does not come in rontact with the barrel wall-Le., that there is a clearance-is the diminution of the flow in the channel. This arises from the fact that with a clearance there is a net transverse flow and c z l - J = 1 - 6/h # 1
(28)
,4rigorous calculation of parameter c based on equating the transverse flow in the channel to the flow in the x direction in the clearance yields 3
(1 - J ) - J l ,UL
c =
rDa en tan @
for which Equation 25 is an excellent approximation for usual screw geometries and operating conditions. For the very great majority of cases the total flow in a melt extruder can be given by the first term of Equation 26, the second term being negligible by comparison.
Q=
nsrDNhitln - e ) cos2 2
+( 1 - a - J )
If we let No eccentricity factor is assumed in this calculation. The pressure gradient in the X direction over the land may be computed by taking the components of the pressure gradients over the lands in the x and z directions.
(30) J = -6
h
and assume for the moment that -J
CSI
(25)
Equation 23 can be further simplified. __ dP
dz X sin
+
6~aDiV cos + X eh2
=
- -
Q = n r D N h ( t / n - e ) cos2 + ( 1 - a - J ) -
2 6 p r D N a cos hZ
+ sin +
When Equation 21 is substituted in Equation 20, the final expression for flow between the land and the barrel is ,QB
2
= - 6 r D N cos 4 h2
- ~ 2 8 ~ r D Ncos2 = 2h2
+ -p x PL
PL
- X u + - J2rD eJ2 bn tan
+
X a]
(26)
This is a more rigorous calculation of total flow than that derived by Carley and others ( Z ) , based on the approach outlined in Equation 1, which is presented for comparison. nrDNh(t/n
Q=
- e ) cosz + (1 2
-
a)
-
Thus the major effect of clearance is the diminution of flow in the channel, which is indicated in Equation 30 by the fact that J = 6/'h > 0. One side light of interest can be obtained from the expressions for the pressure gradients in the channel. If the components of these pressure gradients perpendicular to the axis of the screw are combined and multiplied by the channel width in that direction, the magnitude of the pressure fluctuation measured by a pressure gage installed in the barrel would be calculated. The pressure rise across the channel from A to G (see Figure 4) can be calculated by using the pressure gradients expressed in Equations 16 and 17. _ -d P
dP' du
A--0
dX1A-B
sin
+ + dP cos +=
++
6prDNc sin @ X sin h2 6psrDNa C O S - cos h2
+x
+=
6 p r D N [c sins + + a h2
The equation for the flow over the lands has two features of interest. First, this flow does not depend strongly on the width of the flight, e. This is perhaps unexpected, but not entirely inexplicable. The pressure gradient across the flight would be decreased by increasing the flight width, but the slit width would be enlarged proportionately, and so changing flight width would have almost no effect on leakage flow. The second point of interest is that when there is no pressure gradient along the screw ( a =
768
Two major differences are evident. First, there is a difference in the second terms (the leakage flow or the amount of material actually passing over the lands). For comparison of the results from the two approaches, the results of two numerical calculations for a typical case are presented in Figure 5. Marked differences between the equations exist a t both high and low degrees of back flow. Marked as these differences are, they are small compared to the differences in
INDUSTRIAL AND ENGINEERING CHEMISTRY
(29)
cos2
+I
(31)
$( 1-en/t)
n
The change in pressure from G to A continuing over the land is the negative of AP,-,. A numerical example of this pressure
SCREW E X T R U S I O N fluctuation is shown in Figure 6 for a typical set of extrusion conditions. The pressure increases relatively slowly as the channel passes under the pressuremeasuring point, and then drops rather rapidly as the flight passes under that point. Such a pattern has been observed experimentally during extrusion of thermoplastic resins. One potential use for this expression of the pressure fluctuation at one point on the barrel is in the estimation of the viscosity in the channel. If the assumption is approximately correct that leakage over the land is small, the magnitude of the pressure fluctuation is directly proportional to the viscosity in the channel. This approach has not been useful in the past, because the pressure gages used have not been dynamically fast enough to measure the magnitude of the pressure fluctuations accurately at the screw speeds required to obtain measurable pressure fluctuations. Power Requirement in Melt Extrusion I n the preceding section the flow of fluid both in the channel and over the top of the lands has been expressed mathematically in terms of the velocity distributions in the two mutually perpendicular directions, x and z, as functions of position on the third mutually perpendicular distance coordinate, y. Such an expression of both these velocity distributions is essential to the calculation of the power requirement of a melt extruder. Because the forces on the fluid in the channel differ markedly from the forces on the fluid between the land and the barrel, the power requirements of the fluid in the channel, Z I , and of the fluid passing over the lands, Z Z ,are computed separately. The total power required, Z , is the sum of the two parts. The power requirements are computed as though the screw were stationary and the barrel turning; the requirements are identical for a rotating screw in a stationary barrel. The previous derivation of the power requirements of the melt in the screw channel by Mallouk and McKelvey (8) considered only the shear rate at the melt surface in the direction of the channel. I t also assumed that the force on the melt surface was in the direction of motion of the barrel surface. I t appears that the shear rates in both the x and z directions at the melt surface must be considered, and calculations based on this more complete picture of shear rates and stresses a t the melt surface indicate that the direction of the resultant force is not necessarily in the direction of motion of the barrel surface. The previous derivation does
0 = 2 in. 0.100 in. e = 0.25 in.
N = 3 0 RPM n= I t=2in. p = I@ p o i s e s
h
v)
Q
Y
+400
-
6 =I 7 . 7 O
a J
0
I
Scr ew
2 Re v o I u t I o n s
3
Figure 6. Pressure increases slowly as the channel passes under the pressuremeasuring point, then drops rapidly
not correctly describe the physical situation. Figure 7 shows the “unwrapped” channel with the flattened barrel surface moving above a t an angle to the lands. The forces acting on the top surface of the melt are vectorially represented. The velocity profiles in the x and z directions are known, and differentiation of each expression permits calculation of the shear rates in the two directions. Multiplication of these shear rates by the viscosity and by the differential area gives the two differential forces, dF, and dF,, which must be acting on the surface of the melt. The vector sum of these two forces gives the resultant force, dF, which must be acting on the surface of the melt to maintain the appropriate velocities and shear rates. This force on the surface of the melt is equal in direction and magnitude to the
+
force that must be exerted on the barrel surface, because the barrel is rigid and is moving a t constant velocity. Calculation of the power input to the system requires multiplication of the velocity by the force on the barrel acting in the direction of motion. The component of dF in the direction of U, dFu, must therefore be taken. The component of dF in the direction of the screw axis is a t right angles to the direction of motion and so no power is consumed in maintaining this force; this force acts as thrust on the screw. The derivation of the expression of the power requirement in the channel, dZ1, using the expressions for velocity given in Equations 6 and 9, is presented below. dZi = U d F u = U [dFv cos 0
+ dFT sin 01
(33)
Figure 7. Forces on fluid at fluid-barrel interface are vectorially represented VOL. 51, NO. 6
JUNE 1959
769
For the usual case in which J is a small fraction, terms in J 2 become negligible, and
$ [ ( l - 3a) + + 3 ~ ) d A (34)
V GaH]l~,l dA = '(1 h
(1
+ 3c)dA
(35)
The combination of Equations 33, 34, and 35 gives dZ1 =
u
[P- (1 + 3a) cos + + (1
h
+ 3c) sin +
1
dA (36)
Noting the definition of U , V, and T (Equations 11 and 12), using Equation 25, and choosing the differential area
- tan -]& en +
d A = [D .
r3D3M2p(1- e n / t ) h
+ 3a) cos24 + (4 - 3 J ) sin2+]dX (38)
In the case in which the land width is negligible compared to the circumference, the above expression simplifies to dZ1 =
r3D3X2,u
[(l h (4
'
+ 3a) cos2 + +
1
- 3 J ) sin2+
dX
(39)
Identical expressions for the power requirement in the channel have been derived by considering only the fluid in the channel and calculating the sum of the terms for viscous heat generation and pressure rise. The power consumed by the liquid in the clearance between the land and the barrel surface may be calculated by the procedure used above. The shear rate at the barrel surface is the sum of the shear rates due to drag flow and to pressure flow in the direction of U. This shear rate is multiplied by the viscosity, the differential area, and the velocity of the barrel to obtain the power input. The derivation involves the calculation of pressure gradients and shear rates in the direction of the velocity U; the calculation is analogous to the derivation above and so the results alone are presented.
TD sin 4 cos ne
770
+a
+ 3a) cos2 + + ( 4 - 3J)sin2+
1
r2D2iVe,UL n dX 6 tan 6
the expression for power in the channel may be written
[(I
[(l
=
r D ( 1 - en/t)dh (37)
dZi =
This latter expression neglects the change in power consumption of the section brought about by the pressure flow through the clearance; the expression is identical to that derived by Mallouk and McKelvey (8). The total power consumed in a differential length of the screw is the sum of the power consumed in the channel and in the clearance, and, if the volume of the land and the effect of pressure flow in the clearance are neglected, may be written from Equations 39 and 41,
- 'k%!%!)]dx e
(40)
dh
+
(42)
T o determine the total power required by a finite length of the screw, the above expression may be integrated with length, taking care of the proper variation of dimensions and fluid properties. In some cases the practical utilization of Equation 42 is limited, in that the magnitude of the viscosity in the channel and over the land is exceptionally difficult to estimate. Such a situation arises when a very viscous non-Newtonian resin is being extruded. In such a case the viscosity varies with the shear rate and hence the screw speed; in addition. the rate of heat generation in the polymer from the viscous shear is large with respect to the rate at which the heat can be conducted from the material. In short, the actual melt viscosity at a point in the channel is uncertain because the melt temperature is not known. In general, the power requirements for the extrusion of thermoplastic resins must be obtained experimentally on equipment geometrically similar to the proposed equipment; scale-up may then be accomplished with the aid of the power equations as indicated by Carley and McKelvey (7). Nomenclature
A B
=
region in screw channel
a
= region above land = fraction of maximum
b
=
D e
= =
f
=
H
=
possible pressure gradient, defines relative amount of pressure flow thread width, measured in direction of screw axis outside diameter of screw land width, measured in direction of screw axis local velocity in A direction of an element in melt at a point (2,
INDUSTRIAL AND ENGINEERING CHEMISTRY
I', 2)
fraction of thread depth = y / h
zi
= thread
depth (measured from root of screw to barrel surface) J = ratio flight clearance: thread depth = 6,lh L = axial length of flighted section of screw n = number of flights in parallel in a multiflight screw A = rotational speed of screw (revolutions per unit time) P = pressure at point (x, y, z ) in channel (assumed independent ofy for given x and z) Q = volumetric rate of discharge of screw Q D = volumetric drag flow in channel Q L = volumetric leakage flow over flights Q p = volumetric pressure flow in channe1 Q z = volumetric flow in direction of channel s = shear stress at melt-barrel interface in the channel s = local velocity in x direction of an element in melt at a point ( X ! Y >2) T = component of U perpendicular to direction of helix e) I = lead of screw t = n(b u = peripheral velocity of barrel, relative to screw (or vice versa), U = DiV u = direction coordinate perpendicular to axis of screw component of U directed along helix u = local velocity in z direction of an element in melt at a point
+
v =
( x , Y, 2)
width of channel measured at right angles to axis of helix w = b cos 9 x = direction perpendicular to depth and axis of helix, positive in T direction Y = direction of depth of channel, varies from 0 at screw surface to h at barrel surface total power required to turn screw z = direction along axis of helix, increasing dieward 6 = clearance (radial) between top of flights and barrel A = direction coordinate parallel to axis of screw, increasing dieward P = viscosity of melt in channel ruL = viscosity of melt in clearance between flight and barrel 9 = helix angle = arctan t / n D z e ' =
z =
Literature Cited (1) Carley, J. F., McKelvey, J. M., IND. ENG.CHEY.45, 989 (1953). (2) Carley, J. F., Mallouk, R. S., McKelvey, J. M., Zbid.,45, 974 (1953). (3) Carley, J. F., Strub, R. A., Zbid.. 45, 970 (1953). (4) Zbid.,p. 978. (5) Gcre, W. L., others, Zbid., 45, 969 (1953). (6) Jepson, C. H., Zbid.,45, 992 (1953). (7) McKelvey, J. M., Ibid.,45, 982 (1953). (8) Mallouk, R. S., McKelvey, J. M., Ibid.,45, 987 (1953). (9) Mohr, W. D., Saxton, R . L., Jepson, C. H., Zbid.,49, 1852 (1957).
RECEIVED for review March 27, 1958 ACCEPTEDMarch 10. 1959
