Extruder channel \
'
\
\\MOVING BARREL \\\ SURFACE
\\
\
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\
By using equations derived ,a relate mixing performance to screw geometry and degree of pressure flow, mixing performance of extruder screws can be calculated and balanced against other requirements, such as output and power consumption
I
\ \
\
\
\ \
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\
\
--STATIONARY CHANNEL
W. D. MOHR, R. L. 5AXTON, and C. H. JEPSON Polychemicals Department, Du Pont Experimental Station, E. I. dg Pont de Nemours & Wilmington, Del.
Co., Inc.,
Theory of Mixing in the Single-Screw Extruder
Tm
SINGLE-SCREW EXTRUDER is widely used as a mixing device in the plastics industry. A common practice is to blend a granular thermoplastic with colorants or other additives, and extrude either the final shape-eg., rod, sheet, or tubing-or ribbon to be cut again into granules. I n the extruder the plastic is melted, and the other component distributed through the viscous liquid. Although diffusion of soluble additives, such as dyes, can assist the mixing process, the effect is not large. This is an example of mixing in a laminar-flow system. A theory, which permits calculation of the goodness of mixing achieved in the single-screw extruder, was developed by combining the theory of mixing in laminar-flow systems (3) with that of melt extrusion (7, 2). The equations developed permit the effects of screw geometry and average residence time to be investigated. The calculated pattern of mixing in the extrudate and the effect of residence time agree qualitatively with experiment. The work demonstrated that this laminar-flow mixing process was susceptible to engineering analysis.
veying force is the viscous drag transmitted to fluid in the channel from the barrel surface, and all the fluid motion derives from it. For simplicity the channel is visualized as uncoiled and laid flat. The barrel surface is considered to move across the channel a t the screw helix angle, +, and with velocity U = nDN. For conyenience, the velocity, U, is resolved into components V and T , parallel and transverse to the channel, respectively. V induces a flow, called the drag flow, QD, having a linear velocity profile (Figure 1,A). This flow is the maximum
pumping capacity of the extruder and does not depend on the viscosity of the fluid. If flow is restricted a t the outlet, however, the pressure builds up, and a pressure gradient is set up in the channel. The effect of this gradient is introduced as a fictitious flow called pressure flow, Q p , which has a parabo1ic"velocity distribution (Figure 1,B). This flow does depend on viscosity. A third occurs in the clearance between the barrel and the flight because of the pressure difference across the flight. This is called leakage flow, QL. The discharge from an extruder is the net volumetric rate of
V 7
DRAG FLOW
"~ 8
P R E S S U R E FLOW
Flow Patterns in Screw Channel In the single-screw extruder, fluid flows through a channel of rectangular cross section. The two sides are the leading and trailing surfaces of the flight, the bottom is the screw root, and the top is the inside surface of the barrel. Fluid is conveyed because of the relative motion of barrel and channel. The con-
C
Figure 1.
Velocity profiles in parallel plane VOL. 49, NO. 1 1
NOVEMBER 1957
1857
Figure 2.
Velocity profile in transverse plane
A NSVERSE FLOW
--Ti-
VELOCITY PROFILE ACROSS A - A
flow across a plane perpendicular to the axis of the screw. This may be calculated as the algebraic sum of drag. pressure, and leakage flow, when these are expressed as rates of flow parallel to the screw’ axis.
Q = Qn - QP - Q L (1 1 Because the leakage flow is ordinarily much smaller than the others, it will not be considered further. In the transverse plane the barrel surface drags liquid toward the leading face
0’
0
I6 14
-
n I1
0.5 ;I
STRIATION THICKNESS, lot MILS l 2 l
II 1 1
I
1;
of the flight. In the absence of leakage, all the liquid must be forced downward to return across the bottom of the channel in a pressure-type flow (Figure 2 ) . This “circulating” movement does not contribute to the transport of liquid along the channel: but is an important factor in the mixing action of the extruder. Mathematical description of these flows is provided by simplified extrusion theory (7, Z), based on these assumptions : The viscosity is the same at all points in the system. This implies a Newtonian fluid and isothermal operation. The width of the channel is large compared to the depth, so that the influence of the sides of the channel on velocity distributions can be neglected. Entrance and exit effects on flow are negligible. ‘The liquid \vets all surfaces and is imcompressible.
Figure 3. Calculated patterns of goodness of mixing over cross section of rod-shaped extrudate
h2 6x (%)
-
L I N E S OF C O N S T A N T O U T P U T PER T U R N /
,
02
/
I
0.4
I
/
06
I
I
l
08
FRACTION OF DRAG F L O W , I - a
+ 3aH2]
The velocitv profile predicted bv this equation is shown in Figure l.C. At any point in the channel h. V , p , and dP,’ dz have finite values; in principle, therefore, a is a determinable quantity. I t is seen to characterize the velocity profile in the plane parallel to the channel. If there is no pressure rise along the channel. dP/dz = 0. and a = 0 . If there is no net flow
Figure 4. Effect of helix angle and pressure flow on maximum striation thickness
1 858
V[(1 - 3a)H
INDUSTRIAL AND ENGINEERING CHEMISTRY
hw
J
0-
simplifies this equation. If, as assumed, leakage flow is negligible, there is no net floir in the transverse plane and c = 1.0. Equation 7 can then be reduced to
The amount of shear received by an element of liquid in the channel was calculated as the sum of the products of shear rate and residence time cor each part of the flow path. Shear in the parallel and transverse planes was calculated separately. Shear received between the end of the screw and the die was approximated by the shear over an unflighted extension of the screw having a
PI
Pl
Qz =
& (2)
Calculation of Shear Received by a Fluid Element
simplifies Equation 2 to u =
c =
(5)
and =
and
(4)
The first term on the right is the drag flow velocity; the second. the pressure floiv velocity. Making the substitutions
(I
(31
Defining
s = T(3H2 - 2 H ) (9) The velocity profile in the transverse plane is shown in Figure 2. The path of an element of fluid is thus a helix of oval cross section, in which the helix angle undergoes a cyclical change around each turn. The effect of increased pressure flow is to reduce the helix angle; in its passage through the channel a n element travels around its loop in the transverse plane more often the higher the ratio: QP”QD = a . Consequently, increasing the average residence time by restricting floiv at the die has a specific effect: The amount of shear contributed by this circulation in the transverse plane is increased, while shear in the parallel plane is not changed.
The velocity down the channel may be expressed (7: 2) as
R E D U C E D RAD1 US
from which Tvhen Qa = 0. a = 1.0. I t appears that a = 0 for drag flow alone and 1.0 for no net flow (pressure flow equal to drag flow), and the floiv is linear in a for intermediate conditions I t can be shown that a = Q P , QD. The velocity profile in the transberse plane has not been formulated before. It may be written by analogy with that in the parallel plane
vdH = hw
Jo-V [ ( I - 3a)H + 3aH2))dH = whV - (1 - a ) 2
=
0
(G)
ENGINEERING ASPECTS OF POLYMER PROCESSES
Figure 5. Pattern of mixing over cross section of extruded rod
length equal to the screw diameter. The shear imposed by the die was calculated. All shear contributions were resolved into components parallel to the screw axis (axial shear), and perpendicular to it (circumferential shear). Finally, these components were corrected for the change in shape experienced in forming the rod-shaped extrudate ; the amounts of axial shear and circumferential shear were summed. and the res-ltant amount of shear, M , for the rlement in the extrudate was obtained by vector addition. Lastly, the striation thickness was ccmputed by means of the equation (3) r = bI X L b (10) MYb Pa In principle. the final position of an element in a laminar-flow process can be predicted from the initial position and the geometry of the system. I n these calculations the starting height in the channel, HI, determined the radial position in the extruded rod. Shear in the channel was calculated for a number of starting heights, and the results were obtained as striation thickness us. reduced radius.
Results of Calculations The equations developed permitted calculation of the amount of shear imposed on a n element of liquid as a function of its initial position in the channel. They were used to investigate the effects of screw geometry and average residence
Figure 6. A.
Cross section o f extruded r o d B.
a = 0.17
time on the mixing performance of the single-screw extruder. Striation thickness in the extrudate was computed for eleven reduced radii, corresponding to eleven initial heights in the channel. The value of H I giving the minimum shear in the extrudate was also found. The assumed conditions and the values of the variables investigated are listed in Tables I and 11.
Table I. Operating Conditions in Computation of Extruder Mixing Performance Inches Minor component Cube side length, l b 0.125 Volume fraction, Yb 0.1 Screw dimensions Length, L 24.0 Diameter, D 2.0 Length of extension, L E 2.0 Die dimensions Land length, LD 3.0 Radius, RD 0.25 Extrudate radius, RR 0.25 Table II. Values o f Variables in Computation of Extruder Mixing Performance Helixangle, 9, degrees 10, 17.7, 30, 45 Ratio of pressure flow t o drag flow, a 0 , 0 . 3 3 3 , 0.50, 0.667,0.80,
Thread depth, h, inch
0.90,0.95 0.10, 0.30
= 0.30
I n Figure 3 are shown three typical graphs of striation thickness against reduced radius in the extruded rod. The parameter is the ratio Q p / Q D = a. The mixing is relatively good near the center and edge of the section, poorest a t approximately half the radius. The maximum striation thickness is lowered by a factor of 2.4, and the average by a factor of 5.8 by reducing throughput to 5y0 of the drag flow ( a = 0.95). In Figure 4 the calculated maximum striation thicknesses are plotted against the fraction of drag flow or the fraction of the maximum pumping capacity; the parameter is helix angle. Mixing is improved by reducing helix angle and increasing pressure flow. At constant output per turn, and equal thread depth, a screw with a higher helix angle would be operating a t a higher ratio of pressure flow to drag flow than one of lower helix angle. This does not overcome the improvement in mixing produced by lower helix angles, as shown by the lines of constant output per turn sketched on Figure 4. Increased thread depth is predicted to improve mixing. For all helix angles the effect of thread depth is greatest for 0.50 5 a 5 0.667. The largest difference in goodness of mixing is found a t @ = 4 5 O , a = 0.667, for which the striation thicknesses a t the point of poorest mixing (rrnpx ) are 0.0298 and 0.01 19 inch for h = 0.1 and h = 0.3 inch, respec...
Figure 7.
(I
...
Cross section of extruded rod B. pL3/pa= 1 . 3
c. VOL. 49, NO. 11
p b / p a = 0.14
NOVEMBER 1957
1859
T
Figure 8. Simplified picture of flow in transverse plane
- 91la
=
fl-.,
0
Also
Y
= tan
o
Hence MT =
tively. At the lesser thread depth goodness of mixing falls more rapidly with increasing helix angle. As a result, the effect of thread depth becomes greater the higher the helix angle (Table 111).
Table 111. Effect of Thread Depth on Maximum Striation Thickness rmax.. Inch ( a = 0 ) h = 0.1 h = 0.3
9
10 17.7 30 45
0.0140 0.0235 0.0346 0.0395
0.00895 0.0134 0.0173 0.0188
Y
~ h ~= . 0.1 .
~
ymal-.,
h = 0.3
1.57 1.75 2.00 2.10
Where comparison could be made, the calculated and observed behaviors were in qualitative agreement. The feed material in all experiments was a blend of I/s-inch cubes of natural polyethylene with similar cubes of the minor component, a concentrate of dark pigment in polyethylene. The concentrate was 4.3575 by volume of the blend. The viscosity of the concentrate was varied by using base resins of different viscosity. The blend was extruded as I/s-inch rod. Cross sections of the rod, 20 microns thick, were cut on a microtome, mounted on slides, and photomicrographed. T h e uniformity of distribution of the dark minor component over the area of the sample section gave a qualitative measure of goodness of mixing. The pattern of mixing over the cross section (Figure 5) of I/s-inch rod extruded a t a low value of a, agreed with that predicted. The preservation of this pattern of relatively poor mixing about the mid-radius requires that the liquid follow a smooth course between the end of the screw and the die. The material near the screw root should form the center; that near the barrel, the outside; and the least sheared material from mid-channel should fill the space about the mid-radius. This pattern is not often clearly observed, because flow between the screw end and die is actually less regular than calculated. The action of pressure flow is beneficial. Figure 6 shows the improvement in mixing obtained by increasing a from 0.17 to 0.30. The calculations were carried out for components of equal viscosity, but the effect of differing viscosities could be
1 860
found by multiplying any calculated striation thickness by the ratio: f i b / , u # (see Equation IO). This ratio significantly influences the goodness of mixing achieved for a given set of conditions. Figure 7 shows specimens for which fib/pLawas 10, 1.3, and 0.14, respectively. T o test this approach to mixing in laminar-flow systems quantitatively, a method for measuring striation thickness is needed. Work on this problem is under way.
For simplicity function FT is defined as
which depends only upon H I . So F'T can be computed for a series of values of HI and the corresponding amountszof shear obtained from the equation
Calculations Shear in Transverse Plane. The flow
path in the transverse plane was, approximated by assuming the element to cross the entire width of the channel at one height, HI, then at once to start back on the complementary streamline at Hz (Figure 8). The relationship betw-een complementary streamlines is given by
The amount of shear in the transverse plane is seen to increase with increasing a, the ratio of pressure to drag flow. Shear in Parallel Plane. The average velocity down the channel for an element starting at height H I is computed as the distance traveled in the parallel plane during one cycle in the transverse plane divided by the time for one cycle.
_-_ H,"
- H," = HZ - H3
-
s2
- u2s1 -
s1
(20)
The average shear rate is obtained by differentiating with respect to H I ; the amount of shear i s given by MV
s1
v1s2
(12)
The average velocity of an element is obtained from the distance traveled divided by the time to complete one cycle
sz
s1
Substituting for s from Equation 9 and integrating give
=
1h
do1 ( ) L
dH
81 sin @
s2
The average shear rate is then found
The residence time in the channel of a screw of length L and helix angle @ is
e
= --
VI
L
(15)
sin @
I t can be shown that the term in parenthesis is independent of pressure flow; the calculations are simpler for the case o f a = 0. Inthiscasev = V H and
where V l is the average velocity down the channel of an element starting at height H I . The amount of shear in the transverse plane is given by
I t is convenient to modify this equation by making the shear rate and average velocity dimensionless. I t can be shown that
INDUSTRIAL AND ENGINEERING CHEMISTRY
81 =
Oil, = 0
(Q/QD)
dvl
dvz
dH1 = dHz
=v
I t is convenient to define the function
ENGINEERING ASPECTS OF POLYMER PROCESSES so that the amount of shear in the parallel plane is given by
Mv
=
L Fv
h sin
Like FT, Fv is dimensionless and depends only on H I . Values of these functions are listed in Table IV. M , is seen to be independent of the pressure flow.
Table IV.
Values Functions FV and FT
Fv
Hi 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.500 0.600
-
0.750
17.527 7.880 4.796 3.309 2.441 1.870 1.461 1,149 0.675 0.278 0.446
FT
- 21.679 - 8.527 - 4.171 - 1.983 - 0.648 0.264 0.936 1.458 2.225 2.754 3.155
M u , c = Mv cos @
N
27rT~dLR drR
(27)
Between two streamlines the volumetric flow rate is constant:
Substitution from Equations 27 and 29 into 26 gives the axial multiplying factor
- MT cos @
(25) A (t a A n @ +e) 1- a
Relationship between Shear in ChanThe effect of nel and in Extrudate.
the change in shape of the flow profiles in the channel to the piston-type flow in the extruded rod is evaluated by calcu-
N
h for D
3(Hi
2L " 3 0 %
h
-
- Hi)I2 X
Pi
_
(HI
- H:)
> 0.05, Equations 32 and
34 must
be used. In these calculations Equations 33 and 35 were used only for h = 0.1 inch, Relationship between Position in Channel and i n Extrudate. The veloc-
ity distribution parallel to the axis is assumed to be parabolic with distance from screw root to barrel surface. The liquid leaving the screw is assumed to follow a streamline path to the final position in the extruded rod. The material near the root will then form the center of the rod, and that near the barrel the outside layer. If the forward flow in the channel has a parabolic profile =
k(H1
- H:)
5
Dh2PR
18 RI ( H I - H f ) 2 ( 3 1 )
0.05.
The corrected axial shear is, therefore H: ( 3
MLR= 18 Ri [D -
LhD2pR 2 4 1 - Hi)](Hi
- Hf)'
X
Q VR'R
ROD Volume elements
(36)
Equating the two fractional flows gives the position in the extrudate, p R , in terms of position in the channel, H I :
Shear
CHANNEL
- 2H1)
I n the extrudate, piston-type flow obtains so that the fraction of the total flow that passes between the center and a circle of radius r is
PR =
Figure 9.
(33)
where k is a proportionality constant that depends on screw speed and geometry. The fraction of the total flow passing between the root and a point a t height H I is
= h
Once Fv and Fr have been computed for the desired values of H I , it is only necessary to substitute values of the several variables in turn to obtain the effects of helix angle, a, ratio of pressure flow to drag flow, a, and thread depth, h.
ca)
2 r ~ dyi Dc drR 2LDpg H : ) [ D - 2h(l
f
+ M T sin @
-
M U , R= Mu.c - X
If
> d ~ - -2 r dLR drR Dc dL
-
The amount of circumferential shear is
where dL,/dL and dyl/drR express the effect on axial shear of changes in dimensions experienced by an element of volume in passing from the screw channel to the rod-shaped extrudate. In passing from channel to rod the volume of the annular element remains constant, so that irDc dL dyi
LhDPR 18 R$ ( H i H:)'
(Fv
-
'
The two shear contributions of the channel are resolved into components parallel and perpendicular to the screw axis-Le., to the direction of extrusionas these are the logical directions to consider in the extruded rod. MAC = Mv sin @
lating multipliers for both components of shear. The annular volume elements considered are shown in Figure 9. The radii of these elements lie on the same streamlines. The axial shear in the extrudate is given by
Hi(3 - 2Hi)l"
between
Screw
(38) and
Die.
Liquid in the space between the screw and the die is sheared both axially and circumferentially. The amounts of shear imposed were approximated by shear in the annulus around an unflighted extension of the screw root. VOL. 49, NO. 11
NOVEMBER 1957
1861
'The forward velocity in this annulus is
from which the residence time is computed = 2 p g (1 -
q)
(44)
The axial shear rate is
Combining Equations 44 and 37 gives
from which the amount of axial shear is
Between streamlines the volumetric flow rate is constant, so that
pa = 1 - (1
2TfD
The shape factor is the same as for the channel (Equation 30). Hence the axial shear in the extrudate contributed by the extension is
TD
-
dro = 2 T f R
p;)"2
7R
drR
(45)
(46)
the volume of the annular element remains constant 2770
dyD
d L n = 2 a r ~drR d L R
(47)
Combining Equations 4 6 and 47 gives
M X , E , R=
dLR
dLD
- f. f D
and from Equation 45
As the circumferential shear rate is a D N / h , the amount of circumferential shear in the extrudate is given by the product of shear rate, residence time (Equation 39), and the shape factor
If h / D < 0.05, Dc can be considered equal to D, as in Equations 33 and 35. This approximation was used in the calculations for h = 0.1 inch. Shear in Die. The velocity distribution in the circular die is assumed to be parabolic
The axial shear rate is then
The residence time is
Hence the amount of axial shear is
The effect of the change in shape of the volume elements in forming the extruded rod must be added as the appropriate multipliers
The shear in the extruded rod contributed by the die is, therefore
Summary
The amount of shear received by an element of fluid in the extruder screw channel was calculated for a number of flow paths. These quantities of shear were used to calculate goodness of mixing, expressed by the striation thickness, as a function of reduced radius in the extruded rod. Equations were developed to show the effects of screw design and operating variables on goodness of mixing. Decreased helix angle, increased ratio of pressure flow to drag flow, and increased thread depth were predicted to improve mixing. Where comparison could be made, the calculated results agreed qualitatively with experiment. This agreement supports the concept of shear as the primary mechanism of mixing, and the essential validity of the simplifying assumptions involved in applying the concept. Nomenclature
Even though the diameters may be equal, the positions of a n element in the circular die and in the extruded rod are not the same because of the difference between the velocity profiles. The relation between the two positions can be derived as follows. The fraction of the total flow passing between the center of the tube and a circle of radius r D is
1 862
a
= ratio of pressure flow to drag flow
in plane parallel to channel
J = local velocity in axial direction F = dimensionless function of H h = channel, or thread, depth H = reduced height in channel, y/'h k = proportionality constant between forward velocity and position in channel 1 = cube side length L = length of screw; (with subscript) length in axial direction -If = amount of shear; product of shear rate and time -1- = screw speed, r.p.s. P = pressure Q = local volumetric rate of flow Q = volumetric rate of flow / = striation thickness, average separation of two like interfaces bet\reen two components; (with subscript) radius R = outside radius s = local velocity in plane transverse to channel T = component of barrel surface velocity perpendicular to flight c' = circumferential velocity of extruder screw = aD"V' 61 = local velocity in plane parallel to channel V = component of barrel surface velocity parallel to channel ZL = width of channel x = coordinate axis perpendicular to flight y = height in channel measured perpendicular to screw root; coordinate axis perpendicular to screw root Y = volume fraction z = coordinate axis parallel to channel = screw helix angle p = reduced radius e = residence time f i = viscosity
SUBSCRIPTS major component minor component C channel D die E unflighted screw extension R extruded rod T transverse plane V = parallel plane 1 = initial position in channel, referring to streamline beginning a t this position 2 = referring to streamline complementary to 1 a
b
= = = = = = =
literature Cited (1) Carley, J. F., Mallouk, R.S., McKelvey, J. M., IND.ENG.CHEM.45, 9749 (1953). ( 2 ) Carley, J. F., Strub, R . A., Zbid.,45, 970-3 (1953). (3) Mohr, \$. D.,'Saxton, R. L., Jepson, C. H., Zbid., 49, 1855 ( 1 9 5 7 ) .
RECEIVED for review May 29, 1957 ACCEPTED September 27, 1957 c
= ratio of pressure flow to drag flow
in transverse plane =
&(%)
D = outside diameter of screw; (with subscript) diameter
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Division of Industrial and Engineering Chemistry, Symposium on Engineering Aspects of Polymer Processes and Applications, Joint with Divisions of Paint, Plastics and Printing Ink and Polymer Chemistry, 131st Meeting, ACS, Miami, Fla., April 1957.
