A common type of screw is one in which the pitch is constant b u t whose thread depth decreases uniformly in the rear section and remains constant in the forward or “metering” section. Such a screw is shown in Figure 5 . For this screw p(x) is, of course, constant and independent of A, as is hz(A) in the metering section of the screw. ht(X)for the rear section of the screw varies linearly with A. When these functions are introduced in the general relationship just developed and the necessary integrations and summations are made, Equation 33 is obtained.
if cot
Q =
f + cscp
(0
[& + $1
The effect of using compression ratios in screw design is pointed up by the consideration of a n extruder setup in which the die opening is so large that essentially there is no backward pressure flow. I n this case k is quite large and the term $ / k drops out. Calculation of the delivery reveals t h a t it is larger than t h a t for a screw with a constant channel depth, Lz, along the entire length of the screw. For instance if L, equals LZand h, is twice hp the discharge will be 9.1% greater than the discharge of a screw with a constant channel depth ha. Pressure is developed in the compression section a t the rear, and this causes a forward pressure flow which raises the total output.
(33)
LITERATURE CITED
(0
This relationship gives the flow- rate from an extruder consisting of the screw described above and a die whose die constant is k .
(1) Carley, J. F., and Strub, R. A., IND.ENG. CHEM.,45,970 (1953). (2) Purday, H. F. P., “Introduction t o Mechanics of Viscous Flow,” p. 10, New York, Dover Publications, Inc., 1949.
RECEIVED for review October
ACCEPTED March 6, 1963.
21, 1952.
Application of Theory to Design of Screw Extruders T h e pumping efficiency of a melt extruder is the delivered power, Q A P , divided by the total power consumed by the screw. By manipulation of the flow and power equations ( I , 4) it is shown that the efficiency of a melt extruder is a function only of the screw and die dimensions and is independent of the screw speed, the throughput, the rise i n pressure of the melt, and the viscosity of the melt. Formulas are derived for the design of efficient extruders to pump melts whose temperatures must be closely controlled during extrusion. The effects of nine design factors on the principal screw dimensions are investigated numerically for a typical example. In extrusion jobs where the melt could absorb only very little of the mechanical heat, large factors of safety have been used to ensure the thermal protection of the melt. The present method offers the hope of substantial savings in both investment and power costs by making i t possible to find the smalIest extruder of near-optimum efficiency that will safely do the job. J. F. CARLEY AND R . -4.STRUBl Polychemicals Department, E. I . du Pont de Nemours 6% Co., Inc., Wilmington, Del.
IPT
I T S most general form the problem of extruder design might be framed this way: T h a t are the dimensions of the extruder which will do a certain processing job a t the lowest cost? Cost here is used in the broad sense that all items of cost that can be anticipated are considered. A processing job may be defined as converting a particular feed material a t a specified rate of production into a shaped product of good quality. A big item of the cost is the power consumed in turning the screw. I n 10 years of eontinuous service the cost of driving a melt extruder will about equal its purchase price, while a plasticizing extruder will conm m e energy a t ten times t h a t rate. Because construction costs vary so much from maker t o maker and because little data are available relating manufacturing costs to screw dimensions, we have turned our attention to power consumption. We can define .a kind of limited optimum design by answering the question: What are the dimensions of the extruder which will require the least power to do a certain processing job?
of a given stream of melt by an amount AP. The useful power is, therefore, equal t o QAP. This is really the heart of the processing job, although other factors, such as product quality, usually must be considered. The efficiency of the screw is this useful power divided by the total power-that is,
PUiMPIKG EFFICIENCY
Q = kAP/p
(3)
AP = Q p / k
(4)
How is this criterion applied to the design of melt extruders? The useful work done by the screw consists of raising the pressure 1
Present address, % Sulaer Bros., Winterthur, Switzerland.
978
E = QAP/Z
(1)
where 2 is the total power as defined by Mallouk and AIcKelvey (4). The efficiency of a n isothermal melt extruder of given dimensions is independent of the throughput, the rise in pressure of the melt, the viscosity of the melt, and the screw speed. CarIey, illallouk, and McKelvey ( 1 ) show that the throughput of such an extruder is given by
Q =
0rM
-
(6
y) A P / p
(2)
and that the discharge through the die ( = throughput) is given by
Solving Equation 3 for AP
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 5
Extrusion Substituting Equation 4 for AP in Equation 2 and solving for Q
Q
=
kaN/(k
+P +
(5)
Y)
Combining Equations 4 and 5
A p = a f l / ( k -k P -k
(6)
Y)
Mallouk and McKelvey ( 4 ) show that the total power is given by
+
Z = W M N ~ qNAP
(7)
I n these equations the quantities a, 8, k , W , and q are parameters containing only the dimensions of the screw and die. By combining Equations 6 and 7 AP can be eliminated from the power equation.
Multiplying Equation 6 by Fquation 6, the useful work may be expressed as (9)
If Equations 8 and 9 are substituted into Equation 1, the efficiency becomes
Thus, we obtain the rather startling result that the pumping efficiency of an isothermal melt extruder depends only on the dimensions of the screw and die. If power costs are t o be as small as possible, the efficiency must be maximized. But the efficiency is a function of some eight dimensional factors. To maximize it, then, this function must be differentiated with respect to each of the dimensional variables, the derivatives set equal to zero, and the resulting eight equations solved simultaneously to get the desired optimum dimensions. These equations are not linear but are full of polynomials and trigonometric functions, and so far we have obtained no general solution. However, a few tentative criteria have been sifted out of our design calculations. For example, it seems that the leakage should be about one twentieth the output, while the
2c
I5
\ D, in.
back flow and leakage together should be about one half the output. The optimum helix angle is somewhere near 2 2 O , though the differences in efficiency are small between 15' and 30'. T h e land width should be as small as is compatible with the strength and wear requirements.
225: 0
1
\
D, in.
APPROACH TO DESIGN WHEN MELT TEMPERATURE IS LIMITED
A situation that arises frequently in the design of melt extruders develops when the melt is fed to the extruder at a temperature so high that only a slight rise in temperature can be permitted in the extruder without risk of polymer degradation. I n this case, the heat evolved by the shearing action of the screw or some part of it must be transferred out of the extruder. This is done by cooling the barrel or by running the cooling medium into a core in the screw or both. Probably the surface temperature of the barrel and certainly that of the screw should not be lower than the freezing point of the polymer. The distribution of temperature in the thread and the clearance under such conditions is very difficult to calculate. The effects of such factors a s screw speed, flight dimensions, and number of flights on the rate of heat transfer is discussed by Jepson (3). For the purposes of this problem assume that the over-all coefficient of heat transfer is known, that it is constant a t all points along the screw, and t h a t axial variations in the melt temperature do not seriously upset t h e equations derived for isothermal melts. If the bulk temperature rises more than about 10' C. along the screw, the effect on viscosity must be taken into account and a valid average viscosity used. If the over-all coefficient of heat transfer is very small, a large area will be required t o transfer the heat developed. This will force us to use large, slowly turning screws. The problem becomes one of getting the smallest screw and, incidentally, the most efficient screw that will do the job. It is clear that many factors must be fixed to completely define a screw design, even in the relatively simple case where the channel dimensions are constant. The design conditions and the data enable us t o fix some of these if the others are chosen arbitrarily. The reason for this is t h a t the flow equation, the power equation, and the temperature requirements place restrictions on the solution. A power balance for the extruder can be written
151. Diameter. D. in
-0
\o
10
\DIAMETER
or
5
LID I
0
I
May 1953
(Qp+Q,)/Qd
1
IO 20 30 Figure 1. Influence on Extruder Size and Power Consumption of LID Ratio 0
0
0.5
*L 0
I*0
1.5
Figure 2. Effect of Flow Loss on Diameter and Power Consumption
0.5 /Qp
Q,
I .o
0
Figure 3. Effect on Diameter and Power of Leakage to Back Flow Ratio
INDUSTRIAL AND ENGINEERING CHEMISTRY
919
Z = Q A P f AE where A E
-
UAAf
Once having found D, the corresponding values of thread depth, screw speed, clearance, and power can be calculated from the restricting equations. It can be shown that if the square root of w is greater than unity, then Equation 12 has only one positive root. The parameters u, v, and w may be recomputed for new value. of helix angle, land width, and so on, and the new dimensions aid power may be calculated. The results of changing the different factors may then be tabulated or graphed for comparison and a set of factors chosen which will determine an extruder with relittively low power consumption. This procedure was applied t o the design of a melt extruder which was t o deliver 1000 pounds per hour of product at a pressure of 4000 pounds per square inch. The exact viscosity of the melt was not known at the time, so viscosity was investigated as one of the factors. &o, no data were available on the rates of heat transfer in extruders handling this particular material. Thermal conductivity data showed that if the heat was transferred by pure conduction, the over-all coefficient of heat transfer would be about 10 B.t.u./(hour)(square foot)( F.). This value was used in most of the calculations, but the effect of increasing the coefficient was also examined. Figures 1 t o 9 show the effects on screw diameter and power of nine factors.
(11)
the rate a t which internal energy is being supplied t o the melt-Le., the product of throughput. temperature rise of the melt, and its heat capacity a t constant volume U = the over-all coefficient of heat transfer (or a suitable average value) A = the area of the barrel and/or screw core available for cooling At = the temperature difference between the melt and the coolant (or a suitable average value) =
Any consistent set of units may be used. For example, if Q is given in cubic inches per second and P in pounds per square inch, then 2 and AE will be in inch-pounds per second, etc. B y eliminating the screw speed, thread depth, and clearance in the flow and power equations ( 1 , d ) , this power balance can be written in terms of D, the diameter, and several other factors. B y choosing values for the other factors, the remaining equation is in powers of D and can be solved for D by fast numerical methods. Space does not permit more than indicating the approach and presenting the final equation 0 2
- U D - vD-z/3 - w
=
0
(12)
The parameters u,v, and w are positive numbers defined by the equations
Sere>%diameter was chosen as a typical sere?? dimension, and in looking a t these figures, note that changes in D are associated v i t h changes in h, N , and 6. Since the processing job, or Q A P , is fixed, a low value of total power represents a high efficiency, and minimum power will correspond t o maximum efficiency. The nine factors t h a t were considered a s independent variables can he thought of as being measured along mutually perpendicular coordinate axes in a nine-dimensional space. What we have done is t o choose, by preliminary calculation, a set of values which gave sensible dimensions - QAp (I5) and a fairly Iow power consumption. Then, moving out along each of the nine axes in both directions and one-byone, we examined the effects of these nine factors on the primary screw dimensions and power consumption. The central point, which appears in nearly all these graphs, will provide a basis for comparison. It is the point whose coordinates are as follows. L I D 7 10; ( Q P &.+I/& = 0 . 5 ; &L/&P = 0.1: ‘p = 20“; p = 3000 pomes; e = 0.25 inch; d ) D = 0.76: and t h e
(13)
1
where c = the thickness of metal between the root of the screw thread and the core surface, Of course c must be sufficient t o carry the required torque load. The other terms are defined in the first paper of this symposium (9).
20
+
20
-
POWER 0
i
15-
D, in.
l5
or 2, hp.
0
Degrees
20
or 2, hp.
0
l V l r c o r i t y , poises
e, in
30
Figure 4. Influence of Helix Angle on Power and Screw Diameter 980
I
DIAMETER
L 10
D, in.
or 2, hp.
50,
0
0
0
2000
4000
Figure 5. How Power and Diameter Are Virtually Unaffected by Viscosity
oh----0.2
5
0.50
Figure 6. Screw Diameter and Power as Functions of Land Width
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 5
Extrusion allowable rise in melt temperature is 9" F. UAt = 720 B.t.u./ (hour) (square foot), corresponding t o a n allowable temperature difference of 72" F. and a n over-all coefficient of heat transfer of 10 B.t.u./(hour)(square foot)(' F a ) , I n Figure 1 both diameter and power are plotted as functions of the ratio of length t o diameter. The same ordinate scale is used for both quantities, but the units of diameter are inches while those for power are horsepower. (This ordinate scale is common to all the following graphs.) The practical design a r e a is somewhere near LID = 10, and the 20 graph shows POWER that LID very 0 0 s t r o n g l y influ?
I
goes up and this is offset b y a correspondingly lower speed and smaller t h r e a d depth. I n the r a n g e of L I D from 3 t o 30, the power increases by about 10%. I n Figure 2
0-O
10-
and diameter seem to pass through gentle minima near cp = 20' t o 21'. The effect of helix angle on thread depth is also important. Since the backflow is proportional to h3 sin2 p, the rapid increase in the angle function is compensated by a corresponding reduction in the depth of the channel. The required output is maintained by a rise in screw speed. Earlier we found that the efficiency of the screw is independent of the viscosity. The graph of power versus viscosity in Figure 5 is just another way of saying the same thing. The second plot shows that the required diameter is also uninfluenced by variations in viscosity. This is a very useful bit of information since it means that the size of an extruder can be chosen without much regard to the viscosity of the melt. Differences in viscosity are reflected in small changes in thread depth and screw speed, however. Figure 6 indicates how diameter and power change as the land width i s e s from zero to half an inch. These plots are not quite linear because other factors, mainly the clearance, are linked with land width. B u t they show that from the standpoint of design, extruders should be constructed with the narrowest possible lands. Of course they must be strong enough to withstand the buckling stresses caused by side thrust. Also, they must not wear so fast that leakage is greatly increased during the life of the screw.
'd /D 0
I
I
I
'i,
D,
in.
or 2 , hp.
U A t , B.t,u./ ft2, hr I
0
5000
I
I0,OOO
15,000
Figure 8. Effect on Diameter and Power of Allowable Heat Flux per Unit Area
How important is the core of the screw in transferring heat, from the melt? With no core, the diameter would have to be larger in order to make up for the area of the core. As Figure 7 shows, this is actually the case. I n this plot, t h e abscissa is the ratio of the diameter of the core t o the outside diameter of the screw. As the core gets relatively larger, the required diameter drops and the power rises very slightly, while the speed increases to maintain the output. The value of 10 B.t.u./(hour)(square foot)(' F.) for the overall coefficient of heat transfer, as already stated, is a very conservative one. Some spotty experiments on other materials showed that actual coefficients might range from 50 t o 100, and there seemed t o be no reason t o think that these values would not apply to the particular polymer under consideration. Likewise it seemed possible to increase the over-all temperature difference for heat transfer. So we examined the effect on diameter and power of increasing the UAt product, and the results are shown in Figure 8. The graphs show t h a t tremendous reduction in
May 1933
INDUSTRIAL AND ENGINEERING CHEMISTRY
981
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 t h e 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 dements in a typical example of such a n extruder. 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 t o 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 t o rubberlike materials. The object of this investigation was t o 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 t h a t are drastically non-Newtonian.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 5
