Simplified Flow Theory for Screw Extruders The flow behavior of
Ind. Eng. Chem. 1953.45:974-978. Downloaded from pubs.acs.org by IOWA STATE UNIV on 01/08/19. For personal use only.
a viscous liquid in the channel of an extruder screw is shown to flow of behavior viscous liquids between infinite parallel plates, one of be similar to the is and other which the stationary moving. Assuming Newtonian behavior of the a was derived which relates the rate of extrusion and the liquid, differential equation to the screw and die die pressure 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. MAELDER, AND J. M. McKELVEY Polychemicals Department, E. I. da Pont de Nemours & Co., Inc., Wilmington, Del. of this
THE
This equation is the general equation of flow7 for Newtonian Its derivation can be found in most books on the liquids. mechanics of viscous flow—for example (H). In the preceding paper, solutions of this partial differential equation were presented. How'ever, 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 w7alls on the velocity distribution is negligible. In other v7ords, 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 width to depth is ten or larger the error will be less than 10%. Most of the screw's 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.
the literature
preceding paper (1) symposium IN pertaining to the problem of viscous flow in extruders but the of viewed. In this
was re-
more paper development simplified useful flow equations is presented. The symbols and nomenclature used in this paper are defined in the preceding paper (I). 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 arrow7, the relative motions will be the same as those existing in an extruder where the barrel is stationary and the screw7 rotates. Assuming that the liquid w7ets both surfaces, the motion of the barrel drags the viscous liquid along with it, w7hile 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 at the screw surface. There is also a directional factor involved, since the channel is inclined at angle to the direction of motion. Therefore, in computing the flow rate in the channel we break up the velocity into two components: one of these acts directly dowrn the channel, and the other acts at right angles to it. We call the component w7hich acts down the channel drag velocity, and the component which acts at right angles to this transverse velocity. At the end of the channel there is generally a die This sets up a pressure gradient or some other restriction to flow7. down the channel causing a flow in the reverse direction to the drag flow We call this the pressure flow. There is one other flow7 that must be considered. Generally the screw does not fit perfectly inside the barrel. In 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 will 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.
DERIVATION
=
Qd
—
Qp
—
Ql
FLOW THEORY
The differential equation for the one-dimensional velocity distribution is obtained from Equation 2 by setting the second derivative of velocitj7 wdth respect to x equal to zero. Equation 2 then reduces to
By integrating Equation 3 twice, Equation 4, which gives the distribution at any point in the extruder channel, is obtained.
.
Q
OF SIMPLIFIED
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 at the root of the screw where y 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 flow7 velocity. The diagrams in Figure 2 show the velocity profiles of these flows. In 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 flow's 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 wrhen mixing and heat transfer in extruders are considered. =
=
(1)
Neglecting the leakage flow, which is usually a small fraction of the other flows, the local fluid velocity at 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.
As shown in Figure 3, the screw lead is related to the diameter and helix angle by the equation t
—
w
r
+
I
Jo Q
vwdy (zI2
}
-
Vwh
(6)
/
wh3
2
22
n
dy
\dzj
2µ
1-1
=
(5)
0
(S)
(7)
12m X
In Equation 7 the first term of the right side is the drag flow rate and the second term is the pressure flow rate.
y
sin
cos
OPERATION
A-A
SECTION
\d\)
=
U )
CONDITIONS
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 at all points in the screw channel. SURFACE
SCREW SURFACE
.....
l
BARREL SURFACE
t,
e
\
«
Diagram of Screw Channel
Figure I.
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. U
=
nw w
cos =
=
(t
=
ttDN ne)
—
cos
BARREL SURFACE
/
at»V NET FLOW nr*
(8)
cos
e) cos (t/n dz ¿ /sin
:
SCREW SURFACE
Figure 2. Velocity Distributions
(10)
—
0
3’°-
(9)
in Screw
Channel
(11)
=
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 cos /dP\ e) cos2 e) sin nvDNh(t/n nh3(t/n n y 2 —
\d\)
""
12m
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,
(12)
Let us consider the most common special case of Equation 12— a single-flight screw whose land width is small in comparison to pitch of the screw. May 1953
(dP\
12m
UNDER ISOTHERMAL
Equation 14 1) in (
screws
m/w/mm.m
W
—
wDh3 sin2
2
3·°
V
12 gives
"
BARREL
T/
(13)
into Equation
Substituting Equation which applies to the special case of single flight which the thread width is neglected. 13
e=J
wD tan
—
INDUSTRIAL
AND
-
©
Substituting Equation
15
m
-
(¥)
into Equation
12,
the flow equation
becomes
ENGINEERING
CHEMISTRY
975
Q
aN
=
ß
—
Dh(i/n
where
nh\t/n
AP
(16)
e) cos2
—
e)sin
—
cos
12 L a
and
ß are
constants which depend only
on
the dimensions of the
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.
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 calculating 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 flow equation, since the same flow mechanism 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 inspec-
tion of Figure
Q
—
k
The proportionality constant, k, depends
(22) on
the geometry of the
die.
3.
Height
=
8
Length Width
=
e cos
xD/cos
=
Consequently, the leakage flow equation is written
rD8sE Ap 12µß
(17)
COS2
where Ap is the pressure drop from one side of the thread to the other. In Figure 3 it is the difference in pressure from point A to point . E is an eccentricity factor to -which the value 1.2 is assigned. It is more convenient to express the pressure drop, Ap, in terms of the over-all pressure drop of the screw, AP. First, let us calculate the pressure drop, APi, through a length of the channel equal to one turn of the helix.
p__AP___ number of turns 1
tAP L
xD tan L
_
AP
.
.
would represent the difference in pressure from In Figure 3, 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 screw channel is linear, Ap can be calculated from AP simply by multiplying APi by the ratio of the distance along the helical channel from C A to the distance B A. —
.
Ap
-
irD tan
API
—
COS
--
TT.D
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.
COS
ttD tan
cos2
AP
(19)
Substituting Equation 19 into Equation 17 and assigning the value 1.2 to E, the leakage flow equation is TriDW tan
Ql--íoííl-"
AP
7
¡ AP\
(20)
V77
Combining Equations 20 and 16 gives the complete flow equation
* 976
-
>
(f)
-
T
(f)
INDUSTRIAL
(21)
AND
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 viscosity of the extrudate. Another look at Equation 21 shows that if a plot of Q versus AP is made, a straight line with a negative slope results. We call such a line a “screw characteristic.” For a given material and screw, 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.
ENGINEERING
CHEMISTRY
Vol. 45, No. 5
_Extrusion_ The die flow equation shows that a plot of the die pressure 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. versus
with Equation 28. Conversely, this screw will raise a given quantity of melt to the highest pressure, when speed is limited. VARIABLE
DIMENSIONS
CHANNEL
Consider now a more general case in which the dimensions of the screw channel are functions of their position along the axial For this case the helix angle and the length of the screw. channel depth must be expressed as functions of X, and Equation 14 may be written Q
=
()
A/t(X) sin
()
cos
A
where
)]3
-
[sin. gOOL2
0Q
(29)
ir3D3N =
2
12
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.
d\ (30)
Screw and Die Characteristics
Figure 4.
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 (7 0). Solving Equations 14 and 22 simultaneously
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
