I
R. E. COLWELL and K.
R. NICKOLLS
Plastics Division, Monsanto Chemical Co., Springfield, Mass.
The Screw Extruder Thermoplastic melt extruder performance, flow through annuli, Iubrication system performance, and design pumps and plastic profile dies may be computed by the procedure ven. The method is especially useful when strong temperature gradients and/or non-Newtonian material characteristics make previously available procedures invalid I
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This short article presents the problem solved, the general methods solution, and typical results. The complete manuscript, including the mathematical model, graphical integration of fundamental equations, detailed results of computer solution, and detailed discussion of data, and five additional figures, is available for $1.00. Address: Editor, I/EC, 1155 Sixteenth St., N.W., Wa3hington 6, D. C., sending cash, money order, or check payable to American Chemical Society.
WHEN a screw extruder is operated at conditions that promote strong heat generation and when boundary surface temperatures cause substantial temperature gradients in the material, the resulting temperature profile affects the extruder capacity, pressure-developing potential, extrudate thermal history, and amount ofmixing. The simple screw extruder can be represented by a system consisting of two parallel plane surfaces in relative rectilinear motion. Procedures for screw design and performance estimation have been published ( 2 , 4 ) for two limiting cases: isothermal and adiabatic for Newtonian fluid behavior. A method is presented for estimating melt extruder performance when the screw and barrel surfaces are maintained at arbitrary temperatures and fluids depart from Xewtonian behavior in an arbitrary manner. Published geometry factors ( I , 6) based on the isothermal Newtonian model are used to adjust values of the computed average velocity. The method is limited to situations in which the screw length is sufficiently long for a reasonable approach to the steady-state condition. Temperature and velocity profiles across the depth of the channel are computed for nonNewtonian fluids, taking into account heat generation and conduction. These results can be used to design extrusion screws and elucidate the performance of equipment and processes.
Theory
As material is forwarded along the screw channel, heat is generated by viscous Aow and transferred through both
the screw and barrel surfaces. A die at the end of the channel causes a resistance and hence an adverse pressure gradient in the channel. These factors influence the temperature distribution two surfaces. The velocity distribution is determined by the velocity of the screw surface with respect to the barrel surface and by the viscosity distribution between these surfaces. As the fluid viscosity is both temperature- and shear-
dependent, it is necessary to know the temperature and shear stress profiles to obtain the velocity profile. The method of solution used (5) is readilyadaptable to graphical or machine computation. The pertinent equations are:
-
VELOGlTY, inches/ssc. -I
0
I
2
3
4
240
280
390
360
TEMPERATURE,
3A --- CASE CASE 3 C
6
400
440
480
520
'E
Figure 1. Computed velocity and temperature profiles show effect of boundary temperatures and flow resistance for a commercial polystyrene VOL. 51,
NO. 7
JULY 1959
841
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Typical experimental results are shown in Figure 3. The neutral screw curve temperature values are equivalent to those computed using the steady-state theory (case 3A in table) at high levels
1
4376"
REF TO CASE 3A LENGTH
:5 0
Inches
U i = 5 66 inches/
SBC
2 0 --
0'
h J
m \
VI
= 0.25inch =
r a
K
.-c
?I*
I-
I 9
1.0
S,
T,
~
Ti, m
a
O
-1 1000
1
I
2000
3000
= = = = = = =
1.00 (dimensionless for this iystern of units) 0.0192 in lb. force/sec. inch F, 0.090 lb. force sec./sq. inch 9.00 lb. force/sq. inch 400°/F. 450'/F. 1.56 dimensionless 0.0089°/F.-1
AP, p.5.i.
Figure 2. Velocity, temperature, and pressure performance were computed A, Adiabatic, low entering temperature. state.
D.
by four procedures
E. Adiabatic, high entering temperature.
C. Steady
Uhi
No. 1A
Inch/Seo.
1c
Arbitrary isothermal Newtonian
Ta
Case
3A 3c
3D
8
F.
1.89 1.89
450
5.66 5.66 5.66
450 300 200
300
maximum temperature moves toivard the upper surface with increasing p .
Experimental (4)
Combining equations 3 and 4 gives
where
The usual boundary conditions are the temperatures at the two surfaces and the velocity of the moving surface. Velocity and temperature distributions are computed for various pressure gradients, p . A value for the shear stress at the stationary surface, SO,is assumed, the temperature distribution obtained, and the resulting velocity distribution computed. The velocity of the moving surface will probably not equal the given boundary value and one then must assume a neiv value for SOand repeat the computation until satisfactory agreement is obtained. Typical results are shown in Figure 1 for two values of the stationary surface temperature for a commercial polystyrene whose viscosity is approximated by the "power law." The average velocity is zero when p = 52.8 p s i . per inch for case 3A and 64.4 p.s.i. per inch for case 3C. The tendency for plug flow in the region nearer the stationary surface at higher values of p is characteristic of a pseudoplastic fluid. The average temperature does not vary strongly with p and the position of
842
of AP. At lower values of AP the performance tends toward the adiabatic mode, but is modified by substantial flow of heat to the material from the barrel. Computed and experimental values for the drag flow ( A P = 0) velocity are in good agreement, after geometry and temperature correction factors from Figure 4 and (I) have been applied. When the screw was cooled by circulating water through its core, the output was reduced, the maximum A P was increased, the characteristic S-shaped curve was obtained, and the temperature
Computed performance curves are presented in Figure 2 for the analog to a extruder screw 11/2 inches in diameter with a channel 0.25 inch deep pumping a commercial polystyrene melt. Adiabatic performance curves A and B were constructed ( 3 ) for two values of entering melt viscosity. The steady-state curve, C, was computed by the procedure described. Average melt temperatures are noted at various points along the performance curves. D is for a constant viscosity of 0.0971 pound force second per sq. inch.
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2.0
7
1.5 IN. OIA. E X T R U D E R h = 0.25 IN. : 0.008 IN. e 2 0.25 IN.
g
I .5
Figure 3. Experimental performance curves for a commercial polystyrene
INDUSTRIAL A N D ENGINEERING CHEMISTRY
e
= 180,i:i C R 7 5 R P M ,POLYSTYRENE
, / NEUTRAL
I .o
0.5
SCREW
-- COOLED SCREW
0 0
I
2
AP,p s
3 I
X
4
I -
was almost constant over the entire range of outputs. Cooling of the sample at AP = 4000 p.s.i. during collection on the pyrometer needle (low flow rate) probably accounts for the lower value (435’/F.). Computed values for the temperature were between 435 O and 439”/F. for the entire range of i . It was not possible to check the computed values of AP, as only one gage was used on the experimental setup; hence, the length of channel used in developing the measured pressure was not known. In Figure 5 a correction factor, F T , is plotted us. the normalized pressure gradient. The typical output at about 5/50 = l / 3 is about 12Yo less than predicted from the isothermal Newtonian case. This departure is relatively independent of screw geometry speed and boundary temperature, but depends on the material flow behavior. The viscosity data used in these computations were obtained using a long-tube capillary viscometer.
Acknowledgment
The authors are particularly indebted to William Ball and Leon Cooper, Monsanto Applied Mathematics Section, who programmed this problem for machine computation. The helpful comment and discumion with-one of the reviewers, Paul Squires, and permission from the Monsanto Chemical Co. to publish this paper are gratefully acknowledged.
1.4
/
1
1.2 I I
09
0.005
Th
/ 07
0 O6 5
m
/
= 4OO0F, = 156 -a 1T-400)
3 0 0089
0 015
07
06
08
f(T)=e
10
0.9
I I
12
13
14
(To’ / Thl)
Figure 4. The drag flow velocity correction factor shows the effect of boundary temperature, and material viscosity temperature dependence
S
Nomenclature
base of natural logarithms flow velocity correction flow FT = generalized total flow correction factor g = screw land to barrel clearance h = screw channel depth or separation distance between parallel plates J = mechanical equivalent of heat = thermal conductivity of fluid K = pressure gradient in x direction p_ = pressure gradient when UT = 0 po AP = pressure difference e
=
FD
= drag
= shear stress in fluid = temperature (primed tempera-
T
tures are in degrees absolute)
To = temperature of surface where z = o = temperature
T h
u uh -
ED UT
x
2
z
2.2
9 ?le 2 .o
0
x, 1.8
of surface where z=h = velocity in x direction = velocity of moving boundary = average velocity for drag flow = average velocity for total combined flow = coordinate in direction of motion = coordinate perpendicular to plane of parallel plates = distance measured in z direction starting a t screw root or stationary plane = fluid viscosity = viscosity of Auid entering screw in adiabatic case = screw helix angle = dimensionless function of T = dimensionless functions of shear stress
SUBSCRIPTS
I-
5
Figure 5. The total velocity corr e c t i o n factor shows the effect of departure from Newtonian flow behavior and isothermal temperature distribution
14
3 I .2
*
= reference state in viscosity equa-
I
= arbitrary isothermal, Newtonian
tions case literature Cited
(1) Bernhardt, E. C., ed., “Processing of Thermoplastic Materials,)’ Sect. 11. “Extrusion,” Reinholcj, New York, 1959, (2) Carley, J. F., Strub, R. A,, others, IND. ENG.CHEM.45,969-93 (1953). (3) Colwell, R. E., Soc. Plastics Engrs., 13th Annual Technical Conference, Tech. Papers 3, 153-68 (1957). (4) McKelvey, J. M., IND. ENG. CHEM. 46, 660-4 (1954). (5) Prager, W., Rivlin, R. S., Brown University, private communication, 1955. (6) Squires, P. H.,.SPEJournal 14, 24-30 (May 1958).
88
0
02
04
0.6
08
I .o
RECEIVED for review January 19, 1959 ACCEPTEDMay 4, 1959
P/7, 1
., VOL. 5 1 ,
NO. 7
JULY 1959
843
