I
diameter can be composition level, a relatively large temperature rise can be tolermade at a slight ated. This means that some of the shearing work can be abcost in power if sorbed as internal energy, so that the amount of heat that must be better heat transtransferred out is much smaller. Figure 9 shows how drastically f e r c a n b e exthe size of the screw may be reduced if the allowable temperature 20 p e c t e d . As the rise can be increased. If all the heat must be removed and the d i a m e t e r i s rerise is zero, the screw diameter will be nearly 11 inches and the POWER y o o-o-o-o duced, the speed length 110 inches. If the melt temperature can rise 50' F., an rises, in order to extruder of half that size will do, and the increase in power is less maintain the rethan 10%. or 2 , hp. quired output and p r e s s u r e . Since SUMMARY L / D is kept conA limited optimum design is defined as one in which the pumpstant a t a value of ing efficiency is maximum. This efficiency is in all cases indepen10, these smallerdent of melt viscosity, screw speed, throughput, and the pressure d i a m e t e r exdeveloped. A method is outlined for the design of melt extruders are also truders in which the melt temperature is limited and heat must be shorter. transferred from the melt. Finally, the influence of various deAn important sign factors is related to screw dimensions and power requirefactor in the dein a typical example of such an extruder. ments sign of a melt extruder is the perTemperature Rise, OF, LITERATURE CITED m i s s i b l e rise in I , 0 the temperature (1) Carley, J. F., Mallouk, R. S., and McKelvey, J. M., IND. ENG. of the melt as it is CHEM.,45, 974 (1953). pumped through ( 2 ) Carley, J. F., and Strub, R. A . , Ihid., 45, 970 (1953). the machine. If (3) Jepson, C. H., I h i d . , 45, 992 (1953). t h e polymer is (4) Mallouk, R. S., and McKelvey, J. AT., Ibid., 45, 987 (1953). relatively insensitive to temperature or enters a t a temperature much below its deRECEIVED for review October 21, 1953. ACCEPTED March 6, 1953.
Experimental Studies of Melt Extrusion The experimental investigation described here was designed to test the extrusion flow theory that was presented in the preceding papers of this symposium. The investigation consisted of two parts. In the first part extrusion data were obtained for a viscous Newtonian liquid (corn sirup). In the second part a non-Newtonian polymer melt whose flow behavior a t low shear rates is nearly Newtoniaa (polyethylene terephthalate) was studied. The flow data for both these materials are in excellent agreement with calculations made with the theoretical flow equations. Therefore, i t was concluded that these flow equations accurately describe the extrusion flow of Newtonian liquids and of polymer melts that do not deviate greatly from Newtonian behavior. J. M. MCKELVEY Polychemicals Department, E. I . d u Pont de Nemours & Co., Inc., Wilmington, Del.
T
HE theoretical flow equations presented in the preceding
papers of this symposium (3, 3) offer a potential means for designing plastics extrusion equipment which would be a great advance over the empirical methods now in use. The question arises as to whether these equations can be applied to commercial plastics extruders. Some experimental work has been previously reported. I n 1928, Rowell and Finlayson (6) presented data obtained in the extrusion of a soap solution of 0.0111 poise viscosity in a screw which was 0.50 inch in diameter. Their results confirmed the validity of the flow equations, a t least for low viscosity Newtonian liquids. I n 1951, Pigott (6) presented data obtained in a 1-inch diameter extruder. He presented flow rate-pressure data obtained in the extrusion of oils of 0.5- and 1.1-poise viscosity, and his data were in excellent agreement with
982
the theory. His paper also contained some interesting results in which the radical clearance of the screw was varied while all the other dimensions were held constant. Pigott also presented considerable data on the extrusion of rubber stocks. The pressure-flow rate data for the rubber stocks deviated considerably from the linear relation predicted by the theory, indicating that the flow equations must be modified before they can be applied to rubberlike materials. The object of this investigation was to confirm the flow theory for plastic melts in larger equipment. The work reported represents the first step in our program in which a viscous Xewtonian liquid was used and then a polymer melt whose flow behavior is nearly Newtonian. Further experimental studies are planned to investigate polymers that are drastically non-Newtonian.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 5
Extrusion EXTRUSION FLOW BEHAVIOR OF A VISCOUS NEWTONIAN LIQUID
Apparatus and Procedure. Figure 1 shows a diagram of the extruder used for these experiments. This extruder consists of a special 2-inch diameter barrel, constructed of cold-rolled steel (SAE 4140) and mounted on the drive mechanism of a No. 1 Royle extruder. The screw for this extruder is constructed of the same steel as the barrel, and the flights are tipped with Ampco bronze. Table I gives the dimensions of the screw.
-P
Figure 1.
Calculation of Extruder Constants. The extruder flow equation for isothermal operation is written
Q
= QD
- QP - Q L =
0ciV
- p -A/*P -
y-A/*P
(1)
For purposes of this work it is more convenient t o write Equation 1 in a slightly different form, in which the extruder length is not included in the constants p and y . Equation 2 defines the new constants p’ and y’.
The constants a,p’, and y‘ depend only on the dimensions of the screw and can be calculated with either the exact two-dimensional flow equations ( 2 ) or with the simplified one-dimensional equations ( 3 ) . Using both methods values have been calculated for a,p’, and y l , The compression section of this screw has very little effect on the screw characteristics, Therefore, the screw constants were calculated only for the metering section of the screw. The error involved in ignoring the small length of compression section is about 1%. The results of these calculations are shown in Table 111.
Experimental Extruder
TABLE111. CALCULATED SCREWCONSTANTS The barrel is provided with alternating temperature and pressure taps constructed so that the tips of the thermocouples and pressure gages are flush with the inner barrel surface and are in contact with the liquid in the screw channel. The distance between adjacent taps, 1.796 inches, is equal to half the pitch of the screw. With this spacing of the taps, a t any given time, all the pressure gages and all the thermocouples are in the same relative position with respect to the sides of the screw channel (Figure 1). The pressure gages consist of ordinary Bourdon tubes filled with silicone grease and fitted with a greasefilled stem. A metal piston in the tip of the stem transmits the pressure to the grease and minimizes loss of grease from the stem. The construction and operation of this type gage has been described by Carley ( 1 ) .
Constant U
8’ Y’
From Exact Equations 0.786 cu. inch 0.178 X 10-8 inch4
.....
From Simplified Equations 0.856 cu. inch 0.178 X 10-8 inch* 0.0038 X lO-sinch4
For isothermal operation the extruder discharge equation is written (3)
Introducing the new constants p‘ and
yl,
Equation 3 becomes
(4) TABLEI. DIMENSIONS OF EXTRUDER SCREW
Q
Section A Section B Section C Length of section, inches 4.25 5.00 14.75 Outside diameter, inches 1.982 12 8 : . 1.982 Channel depth, inch 0 491 0.116 Radial clearance, inch 0,009 0.009 0.009 Pitch, inches 2,000 2.000 3.593 Axial land width, inch 0.433 0.433 0.433 Helix angle 17.8’ 17.8’ 30’ Channel depth tapers uniformly from 0.491 t o 0.116 inch.
The head of the extruder accommodates interchangeable cylindrical dies and also a pressure gage and a thermocouple which measure the temperature and pressure of the liquid just before it enters the die. Three cylindrical dies were used for these experiments. Table I1 gives the dimensions of these dies.
TABLE 11. DIMENSIONS OF CYLINDRICAL DIES Die No. 1 2 3
Length, Inch 0.5055 0.5073 2.0035
Diameter, Inch 0.0710 0.1251 0.1263
The proportionality constant, m, relates the screw speed to the output of the extruder and depends only on the screw and die constants. Since all the dies used in the experiment8 were cylindrical, the die constant, k , can be calculated from the Poiseuille eauation. (5) where d = diameter of the die and LD = length of the die. Table IV summarizes the results of the calculations of the extruder constants. Free Discharge Flow. With the die removed from the front of the extruder there is no resistance to flow and therefore no net pressure build-up along the screw. I n this case the pressure and leakage flows do not exist, and pure drag flow or free discharge conditions prevail. The free discharge flow rate of the extruder was determined in the first experiments. Table V summarizes these data.
TABLE IV. At the start of a run, the liquid corn sirup (purchased from the Corn Products Sales Co., Philadelphia, Pa.) was poured directly into the feed hopper of the extruder. The drive was adjusted to the desired speed, and temperature and pressure readings were made a t frequent intervals. Discharge rates were determined by weighing timed samples of the liquid discharged from the die.
May 1953
SUMMARY OF CALCULATIONS OF EXTRUDER CONSTANTS
lo X 106,Cu. Inches ma, Cu. Inch m*b, Cu.Inch Die No. 1 1.23 0.077 0.084 0.401 0.436 11.87 2 0.169 3.12 0.184 3 a m waa calculated using values of u and 8’ from the exact equations. b m* was calculated using values of 01 and 8’ from the simplified equations.
INDUSTRIAL AND ENGINEERING CHEMISTRY
983
TABLEV. RESULTS O F FREE DISCHARGE EXPERIMENTS Screw Speed,
Discharge Rate, Grams/Sec.
R.P.S.
Rate Grams/Rev.
1.41
0.091 0.167 0,333 0.500 0,666 0.833
15.5 17.8 17.5 16.7 15.9 16.1
2.98
5.85 8.33 10.62 13 48
Average
16.6
The volumetric discharge rate calculated 4 ith the drag flow equation can be used with the measured weight flow rate to calculate the density of the corn sirup in the screw channel. The calculated rate is 0.786 cubic inch per rev., which when divided into the average measured rate of 16.6 grams per rev. gives a density of 21.1 grams per cubic inch. This value x a s used to convert the measured weight flow rates! to volumetric flow rates in all the following experiments. The calculated density is about 8% lower than the density specified by the manufacturer of the corn sirup. This is due primarily to the formation of small air bubbles and the subsequent expansion of the sirup in the screw channel. The expansion of the sirup makes the direct measurement of density under conditions comparable to those in the extruder impossible and therefore the calculated densitl- is probably the best value. Discharge through Dies. Five runs were made using the three different dies described in the previous section. The data from these runs are summarized in Table VI. In each of these runs the viscosity of the corn sirup was different owing to differences in temperature and (probably) the water content of the sirup. The viscosity of the corn sirup is very sensitive to moisture content and since the sirup was re-used many times the moisture content undoubtedly changed with time. However, Equation 4 shows that, for a given die, the discharge rate should be independent of the viscosity and depend only on the screw speed.
Pressure and Temperature Gradients. During these expeiiments there were three pressure gages in the barrel of the extruder and also a gage in the head. A plot of the pressure readings from these gages versus the distance along the barrel gives a picture of the pressure profile down the length of the screw. Figure 3 is a plot of the pressure profiles obtained during the runs when the corn sirup was being extruded through the dies. Thesc lines have a constant slope which is to be expected for isothermal Newtonian flow in a channel of uniform dimensions. R h c n a viscous - _, liquid passes through a screw extruder, a
32
___
to rise. From the t h e o r e t i c a l potrer equations for screw extruders ( 4 ) t h e amount of power dissipated can be calculated for a liauid of
W
assumed to be perfectly insulated (adiabatic extruder) the m a x i m u m possible rise in temperature of O b h 4 6 b IO 12 I4 16 18 the material can be DISTANCE F R O M D I E (INCHES) calculated. This Figure 3. Pressure Profiles calculation was made for each of the runs listed in Table VI, and the results are shown therein. Most of these values are less than 3" C., but one is as high as 10" C. These calculated values are definitely on the high side, for two reasons: First, the large volume of metal which makes up the barrel of the extruder acts as a very effective conductor of heat back t o the massive drive housing. Here it escapes by convection and radiation. Therefore, the assumption of adiabatic operation made for this calculation is probably not justified. Secondly, the power dissipation calculation was made on the basis of a liquid of constant viscosity. Since there is some rise of temperature of the liquid, there is a corresponding decrease
TABLEVI.
SUYMSRY OF C O R N
SIRUP EXTRGSION DATA pp,
1 .2
Figure 2.
.4 .6 SCREW SPEED
.8 (RPS)
I
1.0
I
1.2
I
1.4
Extruder Flow Characteristics
Figure 2 is a plot of the experimental data. The solid lines show the theoretical extruder flow characteristics calculated from the exact flow equations. The dashed lines show the characteristics calculated from the simplified flow equations. The experimental points are evenly distributed close to the solid lines, indicating that the two dimensional flow equations accurately describe the flow behavior of Newtonian liquids in the screw extruder. The slopes of the lines calculated from the simplified flow equations are about 9% higher than those experimentally observed. The error is due mainly to the fact that the simplified flow equations ignore the effect of the channel walls on the velocity distribution. 984
Lb.-
TDQ, Sec./Sq. C. Inch
Die No.
R.P:S.
Pa Lb./ Sq: Inch
Q Cu. Incil/Sec.
3 3
0.87 1.24
295 320
0.152 0.219
30 33
0.0060 0.004G
2 2 2
0.17 0.32 0.63
35 35 33 33 34 35
0.0056 0.0058
0.32 0.63 0.90 1.24
0.073 0.129 0.256 0.062 0,104 0.148 0.210
35
3 3 3 3
35 63 120 92 147 190 232
2
0.32 0.63 0.90 1.24
59 89 119 161 92 139 180
0.152 0.252 0.359 0.500 0,037 0,043 0.061 0,089
35 36 36 37 36 36 37 38
2
2
2 1 1 1
A'
0.32 0.65 0.90 1.24
N E , Lb.-
Sec./Sq. Inch
. .c
ATb,
C.
5 6 0.4 0.8
0.0056 0.0046 0.0045 0.0040 0.0035
0.0046 0.0043 0.0037 0.0030
0.0046 0.0041 0.0041 0.0038 0.0039 0.0039 0.0035 0.0031
0.0057 0.0048 0.0038 0.0034 0.0045 0.0038
0.0031
1.5 1.3 3.1 3.9 4.5
0.6 1.1
1.6 2 . ~ 19
6.4 8.4 10.9
0.0026 225 temperature a t die. calculated temperature rise of the material as it passes through the extruder, assuming adiabatic operation. 0 Not calculated because pressure measurements along barrel were not made.
1
a TO = b AT =
INDUSTRIAL AND ENG INEERING CHEMISTRY
Vol. 45, No. 5
Extrusion in viscosity. For corn sirup the temperature coefficient of viscosity is quite high. Therefore, since the power dissipated is directly proportional to the viscosity, the actual temperature rise must be less than that calculated. The observed temperature rise along the barrel was of the order of 1O C. Therefore, in these experiments no attempt was made to measure the temperature gradients or to correlate the calculated adiabatic temperature rise with any observed temperatures.
OF SCREWUSEDFOR POLYETHYLENE TABLE VII. DIMENSIONS TEREPHTHALATE EXTRUSION EXPERIMENTS
Metering Section 2 11.0 0.071 1.00 2.090
KO.flights in parallel
Length of section, inches Channel depth, inch Pitch, inch Screw diameter, inches Land width, inch Radial clearance, inch Helix angle
Compression Section 2 11.0
0.164-0.071 (t,apering) 1.00 2.090 0.094 0.005
0.094 0.005 8.66'
8.68'
I
Figure 4.
Screw Used for Polyethylene Terephthalate Extrusion Experiments
Extrusion Viscosity. The viscosity of the material can be calculated in two ways: First, the flow rate-pressure drop data for the cylindrical dies can be used with the Poiseuille equation. Using the Poiseuille equation and these data the author calculated the Poiseuille viscosity, ,up, of the corn sirup for each run. These values are listed in Table VI. The extrusion viscosity, PE, of the material in the screw channel can be calculated from the pressure profile data with the extrusion flow equation. Solving Equation 2 for viscosity
The slope of the pressure profile was measured directly from Figure 3. The calculated values of p~ are also listed in Table VI. The mean difference between the p~ and the p p values is 0.0002 pound (force)-second per square inch or about 5%. Statistical analysis shows that this difference is well within the experimental error. Therefore, we concluded that the effective extrusion viscosity is not significantly different from the Poiseuille viscosity.
multiple orifice dies and a sand pack. A pressure gage, of the type described in the first part of this paper, and a thermocouple were located in the head of the extruder. The screw for this unit has two flights in parallel ( n = 2), and the pitch of the screw is constant over the entire length. The screw has a metering section of constant channel depth and a compression section of decreasing channel depth. Figure 4 is a diagram of this screw showing the dimensions. The dimensions are given in Table VII. During all the runs the temperature of the molten polyethylene terephthalate was maintained a t about 270 O C. The density of this material was measured a t 270" C. in a piston-type rheometer and was found to be 1.184 grams per cc. The screw speed was determined by timing revolutions with a stop watch, and the flow rate was determined by weighing times samples of the extrudate. Calculation of Screw Constants. The screw for this extruder has a constant pitch over the entire length. However, in the compression section the channel depth decreases linearly with length. In the metering section of the screw the channel depth is constant. The integrated flow equation for a screw of this general type has been worked out and was presented in one of the preceding papers of this symposium (3). The flow equation was written nZD2.N cot
Q =
rDap 12 QP
+
May 1953
CSG2cp
k2 + ?I
-d ( h +hz)
[
2 h;h;
(7)
+ $1
Remembering that for the screw used in this work, L1 is equal t o L2, and rearranging Equation 7
EXTRUSION FLOW BEHAVIOR OF A NON-NEWTONIAN PLASTIC MELT
Apparatus and Procedure. Polyethylene terephthalate, which was used in these experiments, is characterized by a rather sharp melting point and a flow behavior that does not deviate greatly from Newtonian. I n this respect it is quite similar t o nylon. For example, a plot of shear stress versus shear rate for both molten polyethylene terephthalate and nylon has only a slight curvature, up to a shear stress of about 40 X lo4 dynes/square cm. Consequently, the viscosity of these materials is relatively independent of the shear rate, and the extrusion flow equations which were derived on the basis of Newtonian flow shouJd apply with little error to these materials. The polyethylene terephthalate used in these experiments was an experimental polymer and is not available commercially. The apparatus consisted essentially of a 2-inch-diameter extruder. The barrel of the extruder is jacketed and the temperature is controlled by circulating Dowtherm vapor through the jacket. With this extruder the polyethylene terephthalate is melted in a separate melting chamber which is blanketed with an inert gas to retard degradation of the polymer. The molten plastic from this chamber then feeds directly into the feed throat of the extruder. The front of the extruder will accommodate any type of die and during these experiments the author used
2
cp
where
fD
=
hi
+ h2 hi + hz
j P
=
hi hi
+ h2
a2 and 0; are the screw constants based on the metering section of the screw, and AP is the pressure rise over the entire length of the screw. The complete flow equation must contain a term for leakage flow and also an area factor. The area factor, J"A, is needed because the flow equations are based on channels of rectangular cross section. In this screw there are fillets in the corners of the channel which reduce the effective cross-section area. The area factor which was used (0.94) is the ratio of the actual cross-section area to the rectangular area that would be available if the fillers were removed. Equation 9 is therefore written,
(9)
Since the ratio of the channel width to depth is relatively small for this screw a serious error would be introduced by using the simplified flow equations. Therefore, the exact two-dimen,sional a2 and 02 were flow equations were used to calculate 012 and 0.; calculated from the following equations:
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
985.
CY2
8n Db2 cos3'p = 79
(lo) 8=1,3,5---
p;
=
nhEb cos 12
2 $ I)-(
p
11 -
(b&)
tanh
L
Figure 6 shows the screw characteristic obtained for each of the three speeds. Within the limits of the experimental error, the slopes of the three characteristics are identical and have a value of -0.072 X in.6 per pound-second. The extrusion viscosity of the molten polyethylene terephthalate can readily be calculated with Equation 13. The calcu(11) lated viscosity is 0.0341 pound (force)-second/square inch (2350 poises).
'1
q=1,3,5---
SUMMARY AND CONCLUSIONS
(12)
The calculated flow equation for the extruder screw becomes
Q
= 0.168-V
-
245.6 X
AP
( 13 )
u
1. One of the conclusions diawn from the extrusion flow theory, for the special case of isothermal extrusion, is that the rate of extrusion, for a given screw and die, is independent of the viscosity of the extrudate and is directly proportional to the screw speed. The flow data for the extrusion of corn sirup shown in Figure 2 confirm this conclusion. These data were obtained for a material whose viscosity varied over a twofold range, yet for a given die all the points fell on the cabulated line. 2. The pressure taps along the barrel of the extruder enable a direct measurement of the pressure profile to be made. The data plotted in Figure 3 confirm the assumption made in the integration of the flow equation, that for isothermal flow in uniform channels, the pressure gradient is constant. 3. Since the drag flow rate in shallow screws is theoretically independent of the viscosity of the materials, the drag flow equations should be applicable to any viscous material, Newtonian or nonIiewtonian, provided that the material wets both the screw surface and the barrel surface. The free discharge flow measurements made with polyethylene terephthalate, shown in Figure 5, are a confirmation that the drag flow equations can be applied to nonNewtonian polymer melts. 4. The slopes of the screw characteristics for polyethylene terephthalate, shown in Figure 6, are constant and independent of the screw speed over the range investigated. However, the die characteristics in Figure 6 show a curvature. This is the type of behavior that is commonly associated with non-Sewtonian liquids. Very often flow rate-pressure drop data for non-NewI
SCREW SPEED ( R P S )
Figure 5 , Drag Flow Discharge Rates Free-Discharge Flow Experiments. B series of runs were made with no die and no restriction to flow on the front of the extruder. The data from these runs are summarized in Table VIII. Figure 5 is a plot of the free-discharge flow rate versus the speed of revolution of the screw. The slope of the straight line drawn through the experimental points is 0.168 cubic inch which agrees exactly with the calculated value. Experiments with Discharge through Dies. Another series of runs was made using three different multiple orifice dies and a sand pack on the front of the extruder. These data are also summarized in Table VIII. Because of the complicated flow behavior of the material through the sand pack it was not possible to calculate the die constants for these runs. However, the pressure-rate data can be used to plot the screw rharacteristics.
OF EXTRUSION DATAFOR TABLE VIII. SUMMARY
POLYETHYLENE TEREPHTHALATE
Screw Speed,
R.P.S. 0.086 0.112 0.141 0.188 0.209 0.233 0.285 0.087
0,192 0.256 0.087 0.192 0.256
(Extrusion temperature, 270' C.) Rate, Rate, Gram/Sec. Cu. Inoh/Sec. 0.283 0.373 0.464 0.615 0,685 0.759 0.937 0.191 0.437 0.584 0.170 0.382 0.526
0 0146 0.0192
0.0239 0.0317 0.0383 0.0391 0.0483 0.00984 0.0228 0 0301 0 00857 0 0197 0.0271
I
I
1
1
Pressure,
Lb./Sq. Inch 0 0 0 0 0 0 0 680 1450 1830 840 1690 2190
I
1000
PRESSURE
Figure 6.
I
1
I
I
I 3000
2000
(PSI)
Screw Characteristics
tonians can be correlated empirically by using a power law in which the pressure is raised to a po.lr-er greater than one, and these data appear to bear this out. However, the linearity of the screw characteristics is difficult to explain. Since the pressure flow in these experiments was only a small part of the total flow it is possible that the measurements vr-ere not accurate enough to detect any possible nonlinearity of the screw characteristic. The scarcity of points and the limited range covered by this data make it impossible to draw any conclusion about this point and further experiments are needed. 5 . On the basis of these experiments we concluded that the flow equations can be applied directly to the business of designing melt extruders (melt pumps) for mildly non-Newtonian melts. LITERATURE CITED
J. F.,I N D . ENG.CHEM., 45, 858 (1953). Carley, J. F , hIslloulr, R, S.. and XoKelvey, J. M.,Ibtd., 4 5 , 9 7 4
(1) Cariey, (2)
(1953).
(3) Carley, J. F., and Strub, R. A., I b i d . , 45, 970 (1963). (4) LMallouk, R. S., and McKelvey, J. hI., Ibid., 45, 987 (1953). (5) Pigott, W. T., Trans. Am. SOC.Mech. Engrs., 73, 947-55 (1961). (6) Rowell, H. S., and Finlayson, D., Engineering, 126, 385 (1928). RECEIVED for review October 21, 1952.
986
I
u
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPrED
March 6, 1953.
Vol. 45, No. 5
