FULLY DEVELOPED FLOW IN SCREW EXTRUDERS. Theoretical and Experimental Study R ,M
. G R I FF1TH,
American Cyanamid Co., Stamford, Corm.
Differential equations are presented for the thermally and hydrodynamically fully developed flow of an incompressible fluid in a metering screw. A method of numerical integration and some results therefrom are described for a "power law" fluid. The numerical results agree with experimental data obtained for comparison.
HE MOTIVATION for studying screw extrusion stems from the Textensive use of the process and its still largely unpredictable behavior. For example, one half of all plastic processed passes through screw extruders, often in a manner as yet only qualitatively described and occasionally with degradation that quantitative understanding might have circumvented. Moreover, screw extruders may be used for pumping, blending, heating, solvent removal, and chemical reaction of doughs, jellies, catalysts, clays, and the like. This article treats only the flow of an incompressible fluid through a screw extruder, with velocity and temperature profiles essentially equal to those in an extruder channel of infinite breadth and length. Also, curvature effects and leakage across the flights are ignored. The idealized flow is believed to be a good approximation to the flow in the final stage of most screw extruders-i.e., those with very long. thin flow channels. The flow described above has been analyzed previously for the isothermal movement of a Newtonian liquid ( 2 , 70, 73) and has been generalized to allow for wall effects ( 7 , 8, 74). Experimental studies (6, 7 7 ) confirm the theoretical work. With neglect of flow transverse to the flights. the analysis has been extended to isothermal movement of pseudoplastic fluid (5) and to nonisothermal flows of fluid of arbitrary rheological
properties ( 4 ) . In the following, the equations describing the nonisothermal flow, including that transverse, are derived for a n incompressible fluid. These are solved in detail for the power law model, and the results are compared with experimental data. Flow of an Incompressible Fluid in a Metering Screw
The helical flow of fluid through a screw extruder is depicted in Figure l . If the channels considered are very thin, earlier work (6) shows that curvature and cross-sectional geometry are probably unimportant; thus for simplicity one unrolls the channel, stretches it out, and treats it as a rectangular duct. Also, for convenience the barrel is considered to turn instead of the worm; this is the situation as it would appear to a n observer on the screw. The flow can now be represented as in Figure 2, where only the projection of the helical streamline on the XJ plane is shown. The channel is assumed sufficiently broad and long to neglect end and edge effects. Therefore, the velocity varies only with level in the channel. The equation of continuity then shows the only finite velocities and velocity gradients to be u y , u ~ , and ~ , u2,=; these are functions only of x . (Subscripts following a comma denote differentiation with respect to the subscript quantity.)
d U
L-2Figure 1 , 180
Screw
Flow in a screw extruder
I&EC FUNDAMENTALS
Figure 2.
Flow in an idealized screw extruder channel
Let the fluids considered be incompressible, isotropic in a state of rest, and with stresses expressible as a power series in velocity and its derivatives. Then for a helical flow, such as is involved here, the stress components have been shown to be (3, 7, 12):
-p -p
pzz =
4 '.
=
+ AI(:Y); P r u + A3(Y)uztr2; pu.
pzu = u y , z d Y ) ;
+ A?(Y)U,,*2
=
-p
=
A4(Y)u,,zuz,z
pz* =
(1)
U,,,?(Y)
(2)
V'Pik = 0
From Equations 1 and 2 it follows that:
+ pzv
z
+
- 0 ; -p,z
pzz.,
= 0
(3)
The variation in the hydrostatic pressure with .r may be ignored; first integrals (of Equation 3 are then: (x
-
XQ)PIU
- U,,.r?
= 0; ( x
- x4)ftz
-
MOMENTUM BALANCE : uV,= = eTG,(x
where the functions A ,and 9 depend only on the square of the average shear rate Y . Frederickson (7) shows how the various functions of Y may be evaluated from viscometric data. If inertial terms are neglected, the momentum balance may be expressed in terms of the stress tensor, p i k , as:
-p.u
where m , n, and b are independent of shear rate and temperature. This expression fails near zero shear rate, but since that occurs only a t zero output from the extruder, it is not a serious objection. The pertinent differential equations taken from the above and written in dimensionless form for the rheology of Equation 11 are:
u z , z 7l
=
0
(4)
- X : ~ ) [ G , ~-( XXQ ) f~
u ~ =, eTG,(x ~
- x 4 ) [ G r 2 ( x- ~
3
BOUNDARY CONDITIONS : x = 0 : uu = us = 0 ; x = 1: x = x3: u y . . = 0 ; x = xp:
G,?(x -
x4)2](1
-
n)/2n
+ Ga2(x- xa)'](l )
~
sin a,ur 0
uy =
(12)
-
n)/2n
= cos a
Uz.= =
ENERGY BALANCE : T.,,[1 - x.=,I - T , ~ x ~ , = ZX,~~=~ -Br e T { [ G V z ( x- ~ 3 ) G12(x ~ - x4)21(n f 1 ) P n - xc,.[Gy*(xe - x 3 ) 2 G?(x, - x4)2](nf 1)/2n)
+ +
BOUNDARY CONDITIONS : x = 1 : T = 0; x = x,:
T.,
=
(13)
0
EQUATION FOR CONJUGATE HALVES OF STREAMLINE:
with boundary conditions : x =
0 : u y = uz = 0 ; x = h:
x = x a : u y . z = 0;
A
= xq:
u y = u2 Uz.=
sin a: ur
= u2 cos a
= 0
(5)
The energy equation is here a balance between heat generated by viscous action and removed by heat conduction. At a point this would be, if temperature changes in the direction are neglected : --kt,zz
(6)
= pzjuji
However, it is assumed that the flow is so rapid that temperature is essentially constant on a streamline-Le., on both the half a t height x and the conjugate half a t x c . It is then possible to satisfy Equation 6 only in the mean; the energy equation becomes :
h: t =
(ti
+
u y Pe
T , u = Br
eT[Gg2(x - x 3 ) 2
f G?(x
- x4)*](n f 1 ) P n
(15)
For a Newtonian fluid with temperature-independent viscosity and operating at zero output, the temperature rise AT per cycle of the helical path would be :
with boundary conditions : x =
Also, in the limit where x = 1 and x, = 0, Equation 14 becomes a n equation from which the dimensionless pressure gradient in the y direction, G,, may be evaluated. Written in dimensionless form, the equations involve four parameters (the helix angle CY,the rheological exponent n, the heat generation parameter Br, and the pressure gradient G z ) ; in dimensioned form h, a,u,,, t i t 2 , k , m , n, b ) . there would be nine quantities (Pz, The assumption of constant temperature on a streamline, inherent in the above treatment, is justified as follows. If all viscous heating went into local heating instead of being conducted to the walls, the energy equation would be :
f t2)/2; x
x,: t , ,
0
$.
Y? I
(8)
Equation 7 is explicitly : -kt,zr
I
D - z
- kt,,, I
=^-Tc
- Y? I
2-=ic
(9)
I t is realistic to take W 5 10 and Pe 2 105, so that AT/Br < 0.01 except for x > 0.98. Since Al'/Br in the x direction is
T h e above treatment of the energy balance is suggested by Kronig and Brink's analysis (9) of extraction from drops and is shown b e h 4 to be realistic. A material balance ir the j direction, for which the net flow is zero, gives a relation tietween x and xc: 0.8
Describing the abovc: idealized flow in a screw extruder devolves then on solving the ordinary, usually nonlinear integrodifferential Equations 4., 9, and 10 subject to the boundary conditions of Equations 5 and 8.
*
0.6
1000 10 I00 0 6 IO
0 13 30' 005 30° I O -0 14 30°
Flow of a Power Law Fluid in a Metering Screw
For the range of materials, shear rates, and temperature prevalent in many extruders, a satisfactory rheological equation is : 7 =
m exp[-bbn(2t
- t,
- t 2 ) / 2 ] Y ( n-
1)/2
(11)
-06
-04
-02
0
02
04
06
08
IO
UZ
Figure 3. velocity
Typical distribution of the z components of
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181
Br -n Pr -
1000 IO 013 100 06 005 I O 10-014
equation of energy, the previous approximation of temperature was used in the exponential term while calculating the new temperatures. Thus. the equation was linear during the successive integrations, and convergence was assured. By special artifices, the program for the above problem could also be used for the limiting cases where viscosity was temperature-independent-i.e.: where b was zero and the equations of energy and motion xvere uncoupled.
8
T(x.07) 443 425 I80
30'
Results of Numerical Study
Computation yielded : the fluid output 1.1s. pressure gradient? the temperature and velocitl- profiles in the channel, the temperature of the output. balues of x 3 and x4, and the derivatives of velocity Z L , , ~ and u , , ~and temperature T , zat the wall, \There x = 1. As a check on the accuracy of the numerical integration, the calculated temperature and velocity gradients were inserted in an energy balance at x = 1. This consistency check is:
x Figure 4.
For 49 runs, the arithmetic average of the differences between the trvo sides of Equation 17 divided by their mean was 4%. Only seven values exceeded 7 % > and the maximum difference \vas 20%. Typical velocity and temperature distributions are presented in Figures 3 and 4. All such data are too voluminous to be presented here but are of considerable interest, since local residence times may be computed therefrom and used together with local temperatures for degradation estimates. T h e flo\v rates found by integrating the velocity distributions for either isothermal or temperature-independent viscosity are presented in Figure 5. T h e temperature gradient at the barrel wall is given in Figure 6 for temperature-independent viscosity. For the machine-calculated points in Figure 6. the ratio of the given temperature gradient to the temperature a t Y = 0.7 is 10.3, with a standard deviation of 0.3. T h e maximum temperature in the flow channel is nearly that a t x = 0.7 and so may be estimated from Figure 6 through the preceding ratio.
Typical temperature distributions
shown below to be of order 1 and since the AT Br given by Equation 16 is usually a n overestimate. it is reasonable to neglect temperature change on a streamline. Solving the Power l a w Equations
I n the general numerical procedure, Br, G,. a? and n were , T were estimated; and approximate chosen; G,: x3. x ~ and velocities were obtained by Runge-Kutta integrations of the momentum equation. Approximate temperatures \vere then obtained by Runge-Kutta integrations of the energy equation using the derived velocities. T h e new approximation to temperature \vas then used in the momentum equation to obtain a second approximation and so on. Csually three trials taking about 2 to 3 hours on a Burroughs 205 lvere required, depending on the accuracy of the initial guesses. I n integrating the
-01-
-0
z0
I
4
3
. i
5
6
I 7
8
9
10
I I1
G, /cosa2
Figure 182
I&EC FUNDAMENTALS
5.
Fluid output vs. pressure gradient for isothermal flow
12
I3
14
15
I
I
I
I
I
1
I
I
I
'7
't
8-
7 L
6-
5-
4-
\
3-
8r--O
2-
02
I-
-a
0
L
I
l
I
I
02
0
Figure 8.
I
I
i
I
2
i 6
I 5
3 G~ / c o s a4p
Fluid output vs. pressure gradient for n = 1
04
02
Figure 6. Temperature gradient at the barrel surface vs. fluid output for b = 0 Lines a r e constructed from Equation 17 using velocity d a t a from Figure points a r e machine computed from Equation 1 3
0
5
7.
1
I
5
4
G,
Figure
0.5
h
U L l2I 2
0
5;
/COS
6
I
7
I 8
,
9
I Br = 100
;O
ap
Fluid output vs. pressure gradient for n = 1.5
-O.' 0
2
I
GZ /cos
Figure
'The fluid output L'S. pressure gradient curves for nonisothermal flo~vare given for various values of n and heat generation parameter in Figures 7 to 10. Figure l l is a typical cross plot of pressure gradient us'. heat generation at constant fluid output prepared from the preceding figures. Though the latter figure is perhaps more convenient for design purposes and more amrnable to extrapolation. the plots in Figures 7 to 10 show the original data. The temperature at :c = 0.7 us. fluid output is presented in Figure 1 2 for various values of n and the heat generation parameter. Figure 13 is a typical cross plot of temperature cs'. heat generation a t constant fluid output prepared from Figure 12. The latter figure s h o w the original data, while the former is more convenivnt for design and extrapolation.
9.
4
3
5
a2
Fluid output vs. pressure gradient for
n = 0.6 Several ratios of use in design are the temperature gradient at the barrel and the average temperature of the output fluid. These parameters are :
'The scatter about the approximate mean values \vas usually less than 207, for the 40 machine calculated points. though somewhat larger values (approximately 16 and 0.8. respectively) Lvould have been better for Br = 1000. VOL.
1
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AUGUST
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183
I
0
1
I
I
I
]
i 0,
I
i
I
0
1 01
I
3t
t
1 -02
I
-0 l o
Figure 12.
1 2
I
I L PL
02
03
04
Fluid temperature vs. fluid flow
_-_I
3
Br
GZ/cosUp
Figure 10.
n=l0
-01
5
Fluid output vs. pressure gradient for n = 0.2
a2
n 1 .s 1.0
0.6 0.2
30' (3
100
10 a2
10'
10'
30'
1000 1000
CY2
V
30' V
0
0
A
10'
X 8
tn"
0
30'
10'
0
A
where is the viscosity in (pounds) (second)/(square inch), du/dr the shear rate in seconds-', and t the fluid temperature in O C. The screw extrusion machine was a 2-inch Hartig No. 908 plastic extruder. The single flighted screw employed had a constant depth and pitch. The dimensions were: CL?
=
30'
D, = 1.996 in. D2 = 2.000 in.
,
I 10
001,
1 1
100
- --
h e
L L
IO00
Br
Figure 11. Pressure gradient vs. heat generation for fluid output = 0.1 (cross plot of data in Figures 1 to 10)
Experimental Test of Theory
To check the practical value of the above work, 1% carboxy vinyl polymer-in-water solution and corn sirup were extruded. The densities of the solutions were measured in Hubbard pycnometers and the rheological properties in a pressurized capillary tube rheometer. At 25" C., the densities of the corn sirup and polymer solution were 1.43 and 1.00 grams per ml., respectively. The rheological data could be fitted by the following viscosity equations within 4Yo and 270, respectively : Polymer:
p = 0.0145(~du/drl)-O.~,26"-31'
Corn sirup: p = 0.0203 184
eO.157 ( 2 6 -
I&EC F U N D A M E N T A L S
t),
21 '-30'
C.
C.
= =
0.128 in. 0.250 in.
Pressures were measured with oil-filled Bourdon gages mounted in the barrel 27.7 and 17.2 inches, respectively, from that side of the feed port nearest the die. The screw extended 0.75 inch beyond the pressure gage. Temperatures were measured with thermocouples 21.5 and 18 inches, respectively, from the feed port. The couples were insulated from the barrel but were mounted flush with its inside surface. During extrusion of carboxy vinyl polymer solution, no effort was made to control the machine's temperature, since viscous heat production was negligible. During corn sirup runs? however. constant-temperature water was circulated in parallel through two 13-inch zones on the 27-inch barrel. Water was also circulated through the screw core except in several runs; in these, approximately 20% reductions in pressure development occurred at zero output of liquid. The theory developed above assumes that temperature is
IO,
~-
Iilcraturc ( I d ) \\ere used. although tlicir validity lor ionisothermal f h v s is untested. I n treating the polymer data! correction factors \vere taken as one?since no theory for these in non-Sewtonian flows \vas available. T h e values of Q Land G, calculated in this manner are plotted in Figure 14 together Lvith the theoretical curves. T h e theoretical curve and experiinenla1 values of G, 2'5. Br, where the latter is computed on the basis of outside diameter. are presented in Figure 15 for zero fluid output. T h e good agreement of theory and experiment is evident; ho\vevcr, a future experimental check over a broader range of Br \\.auld be desirable.
I- 51
S m aL z
'I
Sample Problem ____
03 1 p 0 -
____
.
IC1
I
-~
IC0
-
__
IC00
Br
Figure 13. Fluid temperature vs. heat generation parameter (cross plot of data in Figure 10)
Since the scre\v and barrel surfaces constituie sections of a single streamline, it \vould then be supposed that jacketing one or both Lvalls Ivould produce the same result. 'l'hc difftreni results achieved experimentally lvith and \vithout core cooling indicate the theory is someivhat defective in this one rcs;>ect. Pressure gradients wcre determined by dividing pressure Sage readings by the distance X brtw:en the gage and the feed port. .The axial distance \vas taken to be that between the flights nearest the pi'essure tap and feed port on the feed-end side. 'This distance \vas used instead of the actual value because the axial pressure drop b-nveen flights is zero for z r r o output and is small even for maximum output; therefore, the major pressure rise in the axial direction occurs in jumps iicross the flights. T h e eff-ective distances determined in this \ \ a > ' ~ v c r c23 and 29 inches, or 27 inches on the average. I n calculating Q z and G, for corn sirup data, the curvarurc corrections in the appendix and the \Val1 corrections in the
constan1 on a streamline.
Suppose 400 lb. per hr. of molten acrylic is 10 be extruded i n a 3.5-inch machine with a constant pitch (17,7'). cons~ant depth (0.4 inch). 20-inch scre\v, turning at 120 r.p.m. Find the required die pressure. Let the \\.all temperature be 375' F. and assume a t)-pical, set of material constants: m = 4 (p.s.i.)(sec.)". n = 0.5. 6 = 0.05' F.-'. k = 0.025 (lb.f.)/(sec.)(' F.). p = 1.12 grams per ml. Ignoring the flight thickness and \vall and curvature corrections, one obtains: Q Z cos cy = 0.10, Br = 520. I n the absence of heat generation, one xvould then obtain from Figures 5 and 6 : G, 'cos a = 3 and T(u = 0.7) 'Br = 0.19. Figures 11 and 13 give the corrections necessitated by heat generation, yielding here: G, cos a = 0.57 and T ( x = 0.7) Br = 0.011. These last parameters in dimensional form give the die pressure as 2700 p.s.i. and rhe estimated maximum temperature, that a t x = 0.7, as 489' F. Appendix
Newtonian Flows in Slightly Curved Screw Channels. 'The effect on flow of mild curvat~ireol' the lio\v channel may bc derived analytically for Seivtonian fiows. T h c equation of motion in creeping flow is : y p = p r * u = p grad (div u ) - p curl (curl u ) (18)
0
n
Figure 14,
CORN SYRUP POLYMER tt
0
"
7
1,
I 2 4 RPM
775 49 29 I275
Fluid output flow rate vs. pressure gradient for carboxyl vinyl polymer solution and corn sirup VOL. 1
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185
'J'he divergence of velocity u is zero for incompressible liquids. 'lhus in helical coordinates (75) the p-component of Equalion 1 becomes : fyC,Dr2 COS
CY^
=
+ sin?
4 h21- {cos u.[up,r 4-~ - ' ( 1
O.)Z+]},~
with the correction factors in G', and Oz defined as:
/
, / f ) ? )cos
(24)
II,
Since G, is a constant, Equation 2 may be integrated oncc exactly. 'Then if the trigonometric quantities in CY are treated as constants instead of Tveakly variable functions, a second closed form integration gives the velocity in the p-direction. A third integration gives the flo\v component in the pdirection per unit width:
In the above. the coordinate system was at rest relative to the barrel. Since the screw is in motion relative to this coordinate system. the channel cross section through which the flow of equation 3 passes also has a p component of velocity Lirhich gives a floiv:
Nomenclature n
. I I , .1?. .I3 .
= thermal diffusivity -14 =
equal to Brinkman number for a Newtonian fluid: bmu?"+' k-lhl-n = diameter = axial width of flight = curvature corrections, taken as 1 for 71 # 1 and from Equation 23 and 24 for ti = 1 = correction for wall effect (74) = fd F,: where F, is correction for wall effect
i 74)
= dimensionless pressure gradient iny direction :
=
( & h ) h " -I( mu?n)-l dimensionless pressure gradient in z direction: Aph'+" sin a?(XmfJ,) -'(KDz.\*) -n channel height, ( D I - D ? ) 2 thermal conductivity rheological constants defined by Equation 11 rotational velocity of screw, positive if ciockwise and helix right-handed (facing die end of screw) hydrostatic pressure Peclet number: hu2, 'a stress tensor defined by Equation 1 pressure drop over length X dimensionless floiv in z direction: Q [ p 2 -
=
\ Equation 2 becomes :
Theoretical
Most of the studies on conductivity of heterogeneous mixtures have discussed electrical conductivity ; Brown (3) and de Vries (77) give several references. Even though these discussions apply equally well to thermal conductivity ( 9 , 77, 78), a new description of the theory seems desirable. The usual model of a heterogeneous system consists of a continuous phase (phase 1 ) \vith a discontinuous phase (phase 2) dispersed
(1)
31
for spherical particles. Maxwell’s equation can be obtained by substituting Equation 3 into Equation 1. Thus, when the particles of the dispersed phase are spheres, the theoretical result shows that n depends neither on the size of the particles nor upon the relative size of the conductivities of the two phases. This suggests that other particle shapes should be VOL.
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