Simplified Flow Theory for Screw Extruders T h e flow behavior of a viscous liquid in the channel of an extruder screw is shown to be similar to the flow behavior of viscous liquids between infinite parallel plates, one of which is stationary and the other moving. Assuming Newtonian behavior of the liquid, a differential equation was derived which relates the rate of extrusion and the die pressure t o the screw and die geometry and to the operating variables. Integrated flow equations are given for the special case in which the viscosity of the liquid is constant throughout the screw channel (isothermal extrusion). Equations are also given for the case in which the dimensions of the screw channel are functions of their position along the length of the screw. J. F. CARLEY, R. S. MALLOUK, AND J. B'f. MCICELVEY Polychemicals Department, E. I . d u Pont de il'emours & Co., Inc., Wilmin,gton, Del.
I
N T H E preceding paper ( 1 )of this symposium the literature pertaining to the problem of viscous flow in extruders was reviewed. I n this paper the development of simplified but more useful flow equations is presented. The synibols and nomenclature used in this paper are defined in the preceding paper ( 1 ) . The flow mechanism of the viscous liquid in the helical channel of the screw can be better understood if one imagines that the channel be unrolled and laid out on a flat surface. Figure 1 shows this concept of the screw channel. If the lower plate, representing the screw surface, is held stationary and the upper plate, representing the barrel surface, is moved in the direction of the arrow, the relative motions will be the same as those existing in an extruder where the barrel is stationary and the screw rotates. Assuming that the liquid wets both surfaces, the motion of the barrel drags the viscous liquid along with it, while the stationary plate exerts an equal and opposite drag. The velocity of the liquid, relative to the screw, is a maximum at the barrel surface and zero a t the screw surface. There is also a directional factor involved, since the channel is inclined at angle p to the direction of motion. Therefore, in computing the flow rate in the channel we break u p the velocity into two components: one of these acts directly down the channel, and the other acts a t right angles to it. We call the component which acts down the channel drag velocity, and the component which acts a t right angles to this transverse velocity. At the end of the channel there is generally a die or some other restriction to flow. This sets up a pressure gradient down the channel causing a flow in the reverse direction to the drag flon. Y e call this the pressure flow. There is one other flow that must be considered. Generally the screw does not fit perfectly inside the barrel. I n other words, there is a clearance between the top of the screw threads and the barrel surface. With a pressure gradient along the screw channel, the viscous material Rill tend to leak through this clearance. We call this the leakage flow. The net rate of discharge from the extruder is equal to the algebraic sum of the drag flow, the pressure flow, and the leakage flow. Equation 1 is the flow equation for the extruder, which is really a material balance made under the assumption that the liquid is incompressible.
Q
= QD
- QP - QL
This equation is the general equation of flow for Sewtonian liquids. Its derivation can be found in most books on the mechanics of viscous flow-for example ( 2 ) . I n the preceding paper, solutions of this partial differential equation were presented. However, the resulting flow equations were complicated and difficult to manipulate mathematically. These flow equations can be greatly simplified if the assumption is made that the effect of the channel walls on the velocity distribution is negligible. I n other words, the simplification is made by assuming that the special case of infinite parallel plates is applicable to the problem. This gives us a one-dimensional velocity distribution. The error caused by this approximation is quite small for shallow screw channels. When the ratio of channel Tvidth to depth is ten or larger the error will be less than 10%. Most of the screws that are used for plastics extrusion fall into this category, and consequently the simplified flow equations are very valuable for practical design work and many other applications. DERIVATION OF SIMPLIFIED FLOW THEORY
The differential equation for the one-dimensional velocity distribution is obtained from Equation 2 by setting the second derivative of velocity with respect to 2 equal to zero. Equation 2 then reduces to
(3) By integrating Equat,ion 3 twice, Equation 4,which gives the distribution a t any point in the extruder channel, is obtained. (4)
The constants of integration were evaluated from the boundary conditions. At the barrel surface where y = h, the fluid velocity, relative to the screw, is V , and a t the root of the screw where g = 0, the fluid velocity is zero. The first term of the right side of Equation 4 is the drag flow velocity and the second term is the pressure flow velocity. The diagrams in Figure 2 show the velocity profiles of these flows. I n drag flow the velocity varies linearly across the depth of the channel, while in pressure flow the familiar parabolic distribution is obtained. The addition of these two flows gives the net velocity at each point. It should be remembered that these are the profiles that would be seen on planes parallel to the axis of the screw channel. On the perpendicular planes, only the transverse velocity components, which are essentially closed circular paths, would be seen. The transverse flow is important when mixing and heat transfer in extruders are considered.
(1)
Keglecting the leakage flow, which is usually a small fraction of the other flows, the local fluid velocity a t any point in the screw channel varies with the depth and lateral position of the point. This two-dimensional velocity distribution gives rise to Equation 2.
974
Extrusion The volumetric flow equation can be obtained from the velocity equation by integrating the velocity-area product from the top to the bottom of the screw channel. Equations 5 , 6 , and 7 show how this integration is performed.
=JiT
Q
A8 shown in Figure 3, the screw lead is related to the diameter and helix angle by the equation
t
vwdy
&=
TD tan
(13)
Q
In Equation 7 the first term of the right side is the drag flow rate and the second term is the pressure flow rate.
*
,
---- - ---
A
T2D2Nh sin Q cos 2
Q
- ~ D h sin2 3 Q(:G?) 12P
(14)
OPERATION UNDER ISOTHERMAL CONDITIONS
(7)
A -,-
=
Substituting Equation 13 into Equation 12 gives Equation 14 which applies to the special case of single flight screws ( n = 1) in which the thread width is neglected.
Equation 12 is the basic differential equation of the simplified flow theory. Using this equation as a basis, integrated flow equations may be obtained for various special cases-for example, cases in which the screw dimensions, such as pitch or channel depth, are functions of their position along the screw. Other cases that can be calculated include certain variations of viscosity along the screw. There follow a few of the more useful examples in which isothermal operation is assumed-that is, the temperature of the fluid is assumed to be constant at all points in the screw channel. Consequently, for Newtonian liquids the viscosity of the fluid must be constant a t all points in the screw channel.
-
BARREL
BARRR
SURFACE
y=ht
d
N.V DRAG FLOW
y=o SCREW SURFACE
t-
Y'h
I------'
BPRREL SURFACE
f B
--+
,v.( 3=Or
Equation 7 can be put into a more convenient form if the geometry of the screw thread is considered. Screws that have one or more flights in parallel may also be considered. Figure 3 shows a diagram of a double-flight screw (the simplest multiple-flight screw). If the threads from a section of the screw, which has a length equal to the lead, are unrolled from the root of the screw and laid flat they will appear as shown in bottom part of Figure 3. From the geometry of Figure 3 the following general relations can be established for screws of any number of flights.
V
=
U
COS Q
nw = (t
= TDN COS
- e ) cos
dz = dX/sin
Figure 2.
'p
- e) cos2 - nh3(t/n - e ) sin
nrDATh(t/n 2
Q
1%
Q
cos
Q
NET SCREW SURFACE
Velocity Distributions in Screw Channel
Extruder Flow Equations (Uniform Channel Dimensions). Here all screw dimensions are assumed to be constant over the entire length of the screw. With a uniform channel cross section and a constant viscosity the pressure gradient in the screw channel must be constant. Consequently,
($3 (12)
Let us consider the most common special caae of Equation 12a single-flight screw whose land width is small in comparison to pitch of the screw. May 1953
11
Q
Substituting Equations 8, 10, and 11 into Equation 7 and remembering that there are n flights in parallel the basic equation of the simplified flow theory becomes
&=
B A R R E L SURFACE
N.O.
3-0
(8)
Q
SCREW SURFACE
-
-3-
- ne) cos Q
w = (t/n
P R E S S U R E FLOW
\'.4w
Figure 1. Diagram of Screw Channel
(g) (g) (9) =
=
Substituting Equation 15 into Equation 12, the flow equation becomes
INDUSTRIAL AND ENGINEERING CHEMISTRY
9?5
where
a =
P =
nrrBh(t/ia - e) cos2 q 2 nh3(t/n
- e)sin ____ p cos p 12L
01 and p are constants which depend only on the dimensions of the screw. So far leakage flow has been neglected; this is thought of as a pressure flow through a long narrow slit. For isothermal operation and uniform channel dimensions, the leakage flow is constant at all points across the top of the thread. Therefore, the total leakage flow can be obtained by calculitting the leakage across a length of thread equal to one turn of the helix. The form of the leakage flow equation must be identical to the pressure floa- equation, since the same flo~vmechanism is assumed. Thus, the leakage flow is directly proportional to the width of the slit, directly proportional to the third power of the height of the slit, and inversely proportional to the length of‘ the flow path. The length, width, and height of the slit can be obtained from inspection of Figure 3.
Extruder Operation (Uniform Channel Dimensions). So far only the flow behavior of the fluid in the screw channel has been considered. The operation of the extruder as a whole depends on both the screw and the die. Consider now the flow through the die or filter or whatever is on the front of the extruder. The flow rate of Newtonian liquids through a die of any shape is directly proportional to the pressure drop and inversely proportional to the viscosity of the material. This is shown in Equation 22.
Q
=
IC
(c) I*
The proportionality constant, k , depends on the geometry of the die.
Height = 6 Length = e cos
p
Width = sD/cos
>t=llDXnlp
p
+e
+it+e
4
s
Consequently, the leakage flow equation is written QS =
TDPE A p 12pe cos2 (O
where A p is the pressure drop froin one side of the thread to the other. I n Figure 3 it is the difference in pressure from point A to point B. B is an eccentricity factor to which the value 1.2 is assigned. It is more convenient tu exprem the pressure drop, Ap, in terms of the over-all pressure drop of the scrcw, A P . First, let us calculate the pressure drop, A P , , through a length of the channel equal to one turn of the helix.
In Figure 3, A P l would represent the difference in pressure from point A to point C. For Equation 17, however, the difference in pressure from point A to point B is needed. Since the pressure gradient in the acrew channel is linear, A p can be calculated from AP simply by multiplying A P , by the ratio of the distance along the helical channel from C - A to the distance B - A .
Ap =
-
L cos
(O C O \ ~ (o
L----
AI’
There are now two equations: one describes the flow-pressure relationship in the screw channel and the other describes the flowpressure relationship in the die. The extruder output and pressure must satisfy both of these equations. Consequently, the simultaneous solution of Equations 21 and 22 results in an equation which relates the throughput of an extruder and the physical dimensions of the screw and die.
(19)
Substituting Equation 19 into Equation 17 and assigning the value 1.2 to E,the leakage flow equation is
ir2D2Ptan V, A P = 10MeL
Y
(F)
(20)
Combining Equations 20 and 16 gives the complete flow equation
976
Screw Geometry
p
tan
QL =
Figure 3.
Equation 23 shows that the output of an isothermal extruder with a uniform screw channel is directly proportional to the speed and is independent of the viscositr of the extrudate. Another look a t Equation 21 shous that if a plot of Q versus A P is made, a straight line with a negative slope results. We call such a line a “men- characteristic.” For a given material and sere\\-, each screw speed gives rise to a different characteristic. However, all these characteristics will be parallel and will be separated by a distance which is directly proportional to the screw speed.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 5
Extrusion The die flow equation shows that a plot of the die pressure versus the flow rate will be a straight line passing through the origin. We call this line the “die characteristic.” Figure 4 shows a family of screw characteristics and one die characteristic. The intersection of the die characteristic with the screw characteristic gives the extruder operating point for each speed.
with Equation 28. Conversely, this screw will raise a given quantity of melt to the highest pressure, when speed is limited. VARIABLE CHANNEL DIMENSIONS
Consider now a more general case in which the dimensions of the screw channel are functions of their position along the axial length of the screw. For this case the helix angle and the channel depth must be expressed as functions of A, and Equation 14 may be written
Remembering that the net flow, Q, is constant along the screw length, Equation 29 can be integrated to obtain the total rise in pressure over the length of the screw.
=
x”
dP
A” [
AphL(X) sin V(X) cos &) $ [h(X)I3 [sin p(X)I2
(30)
PRESSURE
Figure 4.
Screw and Die Characteristics
Design for Maximum Output (Uniform Channel Dimensions). Consider the problem of increasing the output of a given extruder. Assume that the barrel and die dimensions cannot be changed and that the speed of the machine cannot be increased. It may still be possible to increase the output if the screw being used is not designed for maximum capacity. The screw dimensions that will give the maximum output under these conditions can be found from the extruder flow equations. Assume that leakage flow can be neglected ( y = 0). Solving Equations 14 and 22 simultaneously =
Ah sin p cos 1 Bh3
+
p
Qp
If AP is substituted for, according to the die flow equation (Equation 22), the integrated equation can be solved for Q. Equation 30 is obtained which gives the delivery of a screw whose helix angle and channel depth vary along the length of‘the screw according to the function &) and h(X).
I n many screws, however, p(X) and h(X) will be different in different sections of the extruder-that is, the screw might have a tapered section, then a section of constant root diameter, then a section of decreasing pitch, and so on. The only difference in
where I.
La---
B = - 7TD 12kL
If the partial derivative of Q is taken with respect to h, set equal to zero, and solved for h
h2
Figure 5.
-
This equation gives the optimum channel depth for maximum throughput expressed as a function of the other dimensions. Similarly, the optimum helix angle may be computed.
Extruder Screw with Compression Section
development for this more general case is that the pressure rise over the entire screw length is the sum of n integrals where n is the number of sections. Consequently, when substituting for AP and solving for Q, a more general relationship is obtained:
Solving Equations 25 and 26 simultaneously the optimum helix angle and channel depth are p
= 30’
(27)
Therefore, to obtain maximum output the screw should be designed with a 30” helix angle and a channel depth as calculated May 1953
I n each section X is taken as a continuous variable and the L’s are the lengths of the various sections. With screws of variable channel dimension, as with constant channel screws, the discharge rate is independent of the pressure rise and the fluid viscosity.
INDUSTRIAL AND ENGINEERING CHEMISTRY
9’11
A common type of screw is one in which the pitch is constant but whose thread depth decreases uniformly in the rear section and remains constant in the forward or “metering” section. Such a screw is shown in Figure 5 . For this screw p(x) is, of course, constant and independent of A, as is hz(A) in the metering section of the screw. ht(X)for the rear section of the screw varies linearly with A. When these functions are introduced in the general relationship just developed and the necessary integrations and summations are made, Equation 33 is obtained.
if cot
Q =
f + cscp
(0
[& + $1
The effect of using compression ratios in screw design is pointed up by the consideration of an extruder setup in which the die opening is so large that essentially there is no backward pressure flow. I n this case k is quite large and the term $ / k drops out. Calculation of the delivery reveals that it is larger than that for a screw with a constant channel depth, Lz, along the entire length of the screw. For instance if L, equals LZand h, is twice hp the discharge will be 9.1% greater than the discharge of a screw with a constant channel depth ha. Pressure is developed in the compression section a t the rear, and this causes a forward pressure flow which raises the total output.
(33)
LITERATURE CITED
(0
This relationship gives the flow- rate from an extruder consisting of the screw described above and a die whose die constant is k .
(1) Carley, J. F., and Strub, R. A., IND.ENG. CHEM.,45,970 (1953). (2) Purday, H. F. P., “Introduction t o Mechanics of Viscous Flow,” p. 10, New York, Dover Publications, Inc., 1949.
RECEIVED for review October
ACCEPTED March 6, 1963.
21, 1952.
Application of Theory to Design of Screw Extruders T h e pumping efficiency of a melt extruder is the delivered power, Q A P , divided by the total power consumed by the screw. By manipulation of the flow and power equations ( I , 4) it is shown that the efficiency of a melt extruder is a function only of the screw and die dimensions and is independent of the screw speed, the throughput, the rise i n pressure of the melt, and the viscosity of the melt. Formulas are derived for the design of efficient extruders to pump melts whose temperatures must be closely controlled during extrusion. The effects of nine design factors on the principal screw dimensions are investigated numerically for a typical example. In extrusion jobs where the melt could absorb only very little of the mechanical heat, large factors of safety have been used to ensure the thermal protection of the melt. The present method offers the hope of substantial savings in both investment and power costs by making i t possible to find the smalIest extruder of near-optimum efficiency that will safely do the job. J. F. CARLEY AND R . -4.STRUBl Polychemicals Department, E. I . du Pont de Nemours 6% Co., Inc., Wilmington, Del.
IPT
I T S most general form the problem of extruder design might be framed this way: T h a t are the dimensions of the extruder which will do a certain processing job a t the lowest cost? Cost here is used in the broad sense that all items of cost that can be anticipated are considered. A processing job may be defined as converting a particular feed material a t a specified rate of production into a shaped product of good quality. A big item of the cost is the power consumed in turning the screw. I n 10 years of eontinuous service the cost of driving a melt extruder will about equal its purchase price, while a plasticizing extruder will conm m e energy a t ten times that rate. Because construction costs vary so much from maker to maker and because little data are available relating manufacturing costs to screw dimensions, we have turned our attention to power consumption. We can define .a kind of limited optimum design by answering the question: What are the dimensions of the extruder which will require the least power to do a certain processing job?
of a given stream of melt by an amount AP. The useful power is, therefore, equal to QAP. This is really the heart of the processing job, although other factors, such as product quality, usually must be considered. The efficiency of the screw is this useful power divided by the total power-that is,
PUiMPIKG EFFICIENCY
Q = kAP/p
(3)
AP = Q p / k
(4)
How is this criterion applied to the design of melt extruders? The useful work done by the screw consists of raising the pressure 1
Present address, % Sulaer Bros., Winterthur, Switzerland.
978
E = QAP/Z
(1)
where 2 is the total power as defined by Mallouk and AIcKelvey (4). The efficiency of an isothermal melt extruder of given dimensions is independent of the throughput, the rise in pressure of the melt, the viscosity of the melt, and the screw speed. CarIey, illallouk, and McKelvey ( 1 ) show that the throughput of such an extruder is given by
Q =
0rM
-
(6
y) A P / p
(2)
and that the discharge through the die ( = throughput) is given by
Solving Equation 3 for AP
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 5
