Power Requirements of Melt Extruders Very little information is available on the power required to drive extruders, and most of what there is consists of statements of the power consumed by extruders in certain specific applications. The theory of viscous flow has been used to study the forces existing in a melt undergoing extrusion. An equation is derived which expresses the power requirements of the melt section of screw extruders in terms of screw dimensions, screw speed, die pressure, and melt viscosity. The total power is the sum of the power consumed in the helical screw channel and that dissipated between the screw lands and the barrel wall. Methods of establishing values for the viscosity parameters involved are discussed briefly. This equation for power consumption is useful in designing melt extruders and in evaluating their performance. R. S. MALLOUK AND J . M. MCKELVEY Polychemicals Department, E . I . d u Pont de Nemours & Co., Inc., Wilmington, Del.
I
N A melt extruder, the liquid enters a t the feed port and
advances along the helical channel of the screw. At the end of the screw, the pressure that has been built up forces the melt through the die. The mechanical energy input from the drive motor to the screw appears in the melt as heat and pressure energy. One of the problems connected with the operation of screw extruders is the calculation of the power consumption. Our approach to this problem is quite similar to the approach used in our derivation ( I ) of the simplified extruder flow equations. On the basis of Newtonian flow a differential equation, applying to shallow flight screws (channel width-to-depth ratio of 10 or greater), has been derived for the power consumption of melt extruders. We have integrated this differential equation for the special case in which the channel dimensions are constant and the extruder operation is isothermal. It is convenient to divide the derivation of the power equation into two parts. In the first part the power, dZ1, used to drag the polymer melt through the heJica1 channel against a particular back pressure is obtained, and in the second part the power, dZ2, which is dissipated in the radial clearance between the top of the thread and the barrel wall is obtained. The sum of these two terms gives the total power. Figure 1 shows the helical channel of the screw unrolled and laid flat. The channel has a depth, h, and a width, w,which is equal to TD Sin 'p. As in the preceding papers, we will again hold the screw stationary and slide the barrel over it, as the relative motions involved will be the same as in an extruder where the barrel is stationary and the screw rotates. The barrel surface moves over the channel with a velocity, U , corresponding to the peripheral speed of the screw. Velocity V is the component of U which acts down the helical channel. Neglect, for the moment, the tops of the threads and consider the length, dz, in which the viscosity of the material is p. The power used in this length is dZ,. dZ1 is equal to the product of the velocity, U , and the force, dF, required to maintain the motion of the channel as shown in Equation 1.
-
dZi = UdF
elemental area over which the stress acts. Therefore, the component of force acting in the direction of the helix, d F 9 , is equal t o the product of the shear stress acting in this direction and the area, dA. This force component can be resolved into the total force, dF, which acts in the direction of revolution of the screw, by dividing by the cosine of the helix angle. This is shown in Equations 2 and 3.
dF9 = SdA dF=-----=dFc cos
SdA cos
(3)
9
For Newtonian liquids the shear stress is equal to the product of the shear rate and the viscosity as shown in Equation 4.
(4) Combining Equations 1, 2, 3, and 4 gives Equation 5.
(5) However, to obtain a useful power formula from Equation 5, the area, dA, and the shear rate ( d v l d y ) must be expressed in terms of the screw dimensions and the operating variables. The shear rate is a variable over the depth of the channel. However, with a rotating barrel and a stationary screw it is the shear rate at the barrel surface which determines the torque required to turn the barrel. I n the preceding paper on extruder flow therory ( I ) we presented an equation relating velocity a t any point in the screw channel t o the depth position in the channel. Therefore. by differentiation we obtain a general equation for the shear rate or velocity gradient, a t any depth. Solving, then, for the shear rate a t the barrel surface, where y = h, Equation 6 is obtained. =
V
h
'il + 2,
(1)
Force is equal to the product of the shear stress and the
9
(2)
Figure 1
987
dP
(x)
The geometry shown in Figure 1 gives the relationship between the area, d A , and the screw dimensions.
d A = wd2
=
T D sin (pdz
(7)
Substituting Equations 6 and 7 into Equation 5 results in dZ1 =
T D U V ~sin h COS 'p
p
dz
+
rDUh sin 01 cos (0
p
dP
(8)
The power formula can be put into a more useful form by expressing the helical length, dz, in terms of the axial screw length, dX, and by expressing the velocities U and V in terms of the rotational speed of the screw as shown in Equations 9, 10, and 11. (9)
What about the case in which there is a variation of viscosity along the length of the screw? Certain special cases can be handled, such as a linear or logarithmic variation of viscosity with length. I n actual practice, however, i t is difficult or impossible to determine the form of the viscosity variation. Therefore, i t is convenient to express both the flow and power equations in terms of the average viscosity, p (Equation 18). f L
-
J,
pdh L
Introducing Equation 18 into Equation 16 gives the power formula based on the average viscosity.
u = -cos -v p The power equation takes the form of Equation 12 when these changes are made. dZ,
=
?r3D3NZp dX ___
h
+
+D2,Vh sin p cos 2 cos? p
p
dP
(12)
Note that the second term of Equation 12 contains all the factors entering into the drag flow formula ( 1 ) . Therefore, by introducing the drag flow rate into Equation 12 its form can be simplified. Equation 13 shows the simplified form which contains the drag flow in its second term.
The power, dZ2, dissipated in the clearance between the top of the thread and the barrel wall, can be calculated in a similar manner. Referring again to Figure 1, the channel depth is no longer h but is equal to the radial clearance, 6, and the width perpendicular t o the helical axis, is e cos p. Neglecting the pressure rise in the clearance, the shear rate is merely V /6. Repeating the same steps but using these new conditions gives the following formula for dZ2.
Since the total power consumed is equal to the sum of dZ1 and The total power consumed over the entire length of the screw is obtained by integrating Equation 15. dZ2, Equation 15 is the final power equation.
Equation 16 gives the power required to turn the barrel when the screw is stationary and vice versa. As one special case let us consider that of a screw which has constant channel dimensions and is operated isothermally. The screw dimensions and viscosity are constant along the length, and the integrated equation can be written immediately.
Equation 17 also shows the effect of pressure on the power requirements. At open discharge (no die on the extruder) the middle term of the equation drops out. If a die is used on the end of the extruder it causes a pressure to be built up, and the power consumption increases. Equation 17 shows that this increase in power is directly proportional to the pressure.
988
Note that the flow equation ( 1 ) can also be written in terms of the average viscosity, /1.
Therefore, if pressure-flow rate data are available the averagc viscosity can readily be calculated. This approach is useful in design work when a definite rate and pressure are specified and is also useful for checking experimental data with the theory. One question that can be raised concerns the viscosity of the material in the radial clearance between the top of the thread and the barrel wall. A few power calculations for machines in which the radial clearance is very small will show that the power dissipated in the clearance may be two or three times as great as that dissipated in the main part of the channel. This power dissipation causes the temperature of the material in the clearance to rise above the main bulk temperature of the material. Consequently the viscosity of the material in the clearance is reduced and the actual povier consumed will be less than the calculated amount. The temperature rise of the material in the clearance depends on several factors, such as the thermal and viscous properties of the polymer, the dimensions of the screw, and the rate a t which heat can be transfezred from the melt through the barrel wall. T o get some idea of the magnitude of the clearance temperature rise we have made some calculations for a typical 2-inch diameter extruder. We have assumed that the extruder is perfectly insulated and that the properties of the material are similar to those of molten polyethylene. Our calculations showed that with radial clearances greater than 8 mils the temperature effect is not too important, the rise in temperature being less than 10" C. With clearances smaller than 8 mils, the temperature rise is much greater-for example, with a 5-mil clearance it is about 40" C. and with a 2-mil clearance it is about 100" C. Remember that these calculations are based on no heat being transferred through the barrel. I n practice some heat can be removed and the temperature rise reduced. We made power calculations for a 2-inch screw with various radial clearances. Correcting for the temperature rise of the material in the clearance, we found that the total power consumption remained just about constant, regardless of the radial clearance. Therefore, the specification of radial clearances should be governed primarily by the allowable temperature that the material can take without suffering thermal degradation, Of course, if the clearance is made too large the increased leakage rate will reduce the capacity of the extruder, and this should also be taken into account. Our present design practice is to make the radial clearance such that the leakage flow is about 10% of the pressure flow. With a 2-inch diameter screw the radial clearance will usually be about 8 to 10 mils and the temperature rise in the clearance is usually quite small. Thrrefore the same value of viscosity can be used for calculating both
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 5
Extrusion the clearance and channel powers. Of course, each design problem is different, and these calculations should be carried out each time a screw is designed or the power for an existing screw is calculated. So far, this discussion has been about Newtonian liquids. Can the power formula also be used for nowNewtonian liquids? There is not much doubt that it can be used for materials such as nylon and polyethylene terephthalate which approximate Newtonian behavior. This statement is based on the fact that the flow equations and the power equations rest on the same basic assumptions and that flow data for the extrusion of polyethylene terephthalate closely verified the extruder flow equations (2). Polymers such as polyethylene are a more difficult problem because the viscosity of polyethylene is influenced to a greater extent by the shear rate. Therefore, in an extruder where the shear
rate in the radial clearance may be ten t o a hundred times greater than in the channel there is a decrease in clearance viscosity. The clearance power term can of course be corrected for this effect if the rheology (shear rate-shear stress data) of the polymer is known. Aside from this correction, it may be possible to use the power formula if an effective viscosity can be specified for the material. Experimental measurements on non-Newtonian materials are needed, however, before this point will be clear. LITERATURE CITED
(1) Carley, J. F,, Mallouk, R. S., and McKelvey, J. M.,
IND. ENG
CHEM.,45, 974 (1953). (2) MoJlelvey, J. M., Ibid., 45, 982 (1953). RECEIVED for review October 21, 1952.
ACCEPTED March 6, 1953.
Extruder Scale-up Theory and Experiments T h e design of extruders receiving solid particle or strip feeds is of great economic interest to the plastics industry. However, no experimentally valid theory of behavior for the solids in the rear of an extruder has as yet been achieved. Using the theory of melt extruders as a guide, together with experimental evidence on the behavior of solids and of “plasticizing” extruders, we have obtained rules for predicting the performance of large extruders from the performance of geometrically similar models. If similar extruders are run adiabatically at equal speeds and one is x times as large as the other, their outputs and power consumptions will be expected to be i n the ratio xz:l, while their output pressures and temperatures will be equal. These rules imply that in many cases the heat of melting can be supplied by the mechanical action of the screw rather than by transfer through the barrel walls. Designs based on this principle should lead to smaller, more economical extruders. J. F. CARLEY AND J. M. MCKELVEY Polychemicals Department, E. I . d u Pont de Nemours & Co., Inc., Wilmington, Del.
T
x
HE preceding papers of this symposium have dealt with the theory and operation of isothermal melt extruders. Of even greater importance to the extrusion industry are so-called plasticizing extruders in which the plastic is fed as solid particles or strips. As has already been shown (3), a large part of the power required by melt extruders is consumed in overcoming the resistance of the viscous material to the shearing action of the screw and barrel. This power takes the form of heat, and if i t is not transferred away, it raises the temperature of the polymer. Now this i8 exactly what is wanted in a plasticizing extruder whose job is to transform the cold feed into a hot, formable melt. But the feed particles are not a melt, and the laws of melt flow cannot be directly applied t o them. On the other hand, by considering those laws rules of similarity for melt can be derived, and there is evidence to show that these scale-up rules apply, approximately at least, to plasticizing extruders too. Thus, they provide a basis for the design of a large extruder when the behavior of a scale model is known. DERIVATION O F SCALE-UP RULES
Let us begin by considering again an isothermal melt extruder. It is not necessary to limit the extruder to one of constant channel section. The simplified flow equations ( 1 ) may be applied to any point along the channel by replacing the quantity A P / L with dP/dh, the pressure gradient a t that point. (If the channel is of constant dimensions, then dP/dX is everywhere the same and May 1953
equal to its average value, APIL.) The flow equation a t any point may then be written
where cy = ?r2D2iVhsin (a cos q / 2 p = ?rDh3sin=Q/12 y = x 2 D W t a n qpllOe
The symbols used in defining a,B, and y are given in a preceding paper (2). Solving Equation 1 for dP/dX and integrating
If the melt is incompressible, the net forward flow, Q, must be the same a t all points, and Q may be taken out of the integrand. It is convenient to assume that the feed enters the extruder at the same pressure that i t has after being discharged from the die Then the die equation may be written
Q
(3)
= kAP/p
If the die is cylindrical, k is given from Poiseuille’s equation as k = ?rR4/8Ld
(4) It can be shown that for dies of other shapes, k will be differently
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
989
