I
E. S. DeHAVEN James River Division, The Dow Chemical Co., Williamsburg, Va.
Extruder Design For u Pseudoplastic Fluid This procedure, especially valuable during preliminary design, permits rapid estimation, by reference to graphs, of single-screw extruder performance with many pseudoplastic fluids
STUDIES
of the action of the singlescrew extruder, from the flow line ( 4 ) and the mixing standpoints (6),have recently been presented, as have applications to semisolid pumping (5). Analyses of the pumping performance of viscosity pumps, as represented by the single-screw extruder, operating with Newtonian fluids have been available for some time ( 2 , 8). Using essentially the same methods, but taking into account decrease in apparent viscosity with rate of shear, these analyses are extended to pseudoplastic fluids. Basic Idealized Analysis
The relation between rate of shear (velocity gradient) and shear stress for Newtonian fluids is -dv/dy = ~ / p . The relation chosen to represent the pseudoplastic fluid,
has the property that either side changes sign algebraically when the sign of T is changed. This is particularly desirable for analyzing viscosity pumps, because, when high pressure differences exist, fluid will flow in opposite directions in different zones of the pumping channel. Under such conditions, if the relation does not have a form much like Equation 1 derivation of the equations for pump analysis will become much more complicated. This form of relation has a further advantage in that, in cases of low shear stress, T , the relation reverts essentially to the Newtonian relation. This is not true of a simple exponential equatione.g., -dv/dy = k T * / ~ ~ L o . For design of a screw viscosity pump, 40% aluminum powder in oil may be represented by Equation 1 satisfactorily up to a viscosity reduction by a factor of 3, and molten polystyrene up to a factor of over 100. The pseudoplasticity constant, C, can be evaluated in Equation 1 by fitting data obtained from concentric cylinder viscometers operated at a number of speeds, or from pressure drop measure-
ments at a number of flow rates through short lengths of tubing ( I , 7, 9 ) . In the case of the concentric cylinder viscometer, Equation 1 is used directly; in the case of pressure drop measurements, Equation 2 is used :
T h e Idealized Model. Consider two parallel plane surfaces, each of breadth ~ e rand length L,, set apart a distance, h. Let the space between them be filled with the fluid and let the edges along the length of the surfaces be sealed in some fashion. Let one surface move a t velocity V, Assuming that the fluid wets the surfaces, the velocity of each fluid element varies with y, the distance from the moving surface. This is illustrated at the left of Figure 1. O n the assumption that steady state has been attained, a force balance, as illustrated at the right of Figure 1, can be written. The resulting equation is: rwL,
+ APyw
= rmwLa
(3 ) (4)
Equations 1 and 4 form the basis for the derivation of relations predicting the pumping performance of the idealized model. The derivation has been presented ( 3 ) , and the numerical evaluations of the derived relations are shown graphically in Figures 2, 3, and 4.
Figure 2 is used to calculate AP,, the pressure gain at shutoff-Le., at zero net flow-of the idealized model as a function of f, the shear rate index. APp is expressed as the ratio to APan, the pressure gain a t shutoff which would be realized with a Newtonian fluid having a viscosity equal to po. The shear rate index,f, was selected by trial during the numerical evaluations as the most convenient parameter for expressing pumping conditions for design purposes. It is defined as follows :
i'+)
POV
The pressure gain at shutoff for a Newtonian fluid has been given by Rowell and Finlayson (8) as follows :
Figure 3 gives AP, the pressure gain of the idealized model, as a function of Q, the net flow rate (or delivery) with f as the parameter. AP is expressed as a ratio to AP,, and Q is expresaed as a ratio to Q j a , the net flow rate at free discharge-4.e.) at zero pressure gain. The free discharge delivery, Qjais giver! by (8):
The power consumed in the idealized model is given in Figure 4. T h e power
STATIONARY PLANE 7
VELOCITY DlSTRlBUTlON
FORCE BALANCE
Figure 1 . From the velocity distribution and force balance, plus the equation for the liquid behavior, the other figures are derived VOL. 51, NO. 7
JULY 1959
813
0.01
0.03
0.10
0.3
f
1.0
3.0
IO
clearance can usually be calculated from Equation 1, substituting V / h for -dv/dy and solving for T . The power consumed at areas of close clearance is simply the production of V , r , and the shearing surface area. The Ineffective Power. I n the screw viscosity pump, power is consumed in the pumping passage between the screw threads because of the ineffective components of velocity in the fluid. This power may be termed the "ineffective power," and it is a power requirement in addition to those discussed thus far. The ineffective power can be calculated from the ineffective components illustrated in Figure 6 by observing that the ineffective components can be analyzed in terms of the idealized model operating at shutoff. The results of such an analysis are:
100
30
Figure 2. Shutoff pressure gain is obtained from the design conditions of pump speed and geometry and of liquid viscosity
fineff (APsnQfd)ineff
is expressed as a ratio to the product of AP, and Q f d , this ratio being shown as a function of the ratio of Q to Q f d withf as the parameter. Finally, Figure 5 , showing the efficiency of the idealized model as a pump, has been included as a guide for design. The same independent variable and paramrtw have been employed as in Figures 3 and 4. Incidentally, the derivation can be extended to the condition of zero speed, giving the flow between stationary parallel plane surfaces :
sidering the screw as stationary and the casing as revolving. The casing wall velocity may be resolved into an effective velocity in the direction of the helix angle and an ineffective velocity perpendicular to it. Figure 6 shows the situation for rectangular threads, to which this discussion is limited. Design calculations are made using the relations for the idealized model, after the following substitutions : q5
t = arctan TD -
(9)
Applications to Design
The basic attack is to assume a pump speed and geometry, to calculate its performance, and to compare this performance with that desired. The assumed pump speed and geometry are modified until the required performance is obtained, approximately. At this point in the calculation, it is usual to design for a nominal flow in addition to the flow rate required from the pump. This nominal flow rate is an allowance to be used later in the calculations for internal and external leakage. The procedure for calculating the performance of the assumed pump, then, consists of first expressing the pump geometry and speed in terms of the idealized model. (Transformations necessary to express screw pump geometry in these terms are discussed below.) Then the shutoff pressure gain is determined from Equations 5 and 6 and Figure 2. The performance (pressure gain us. delivery) is readily determined from Equation 7 and Figure 3. Transformations for the Screw. The action of the screw viscosity pump is basically that of the idealized model. Its action is best understood by con-
814
Power Requirement. The power, Z,, required for the pumping passage itself is readily calculated with the use of Figure 5. The extraneous power consumed in the pump at areas of close
b Figure 3. Pressure gain and i t s variation with delivery are obtained for given design conditions
INDUSTRIAL AND ENGINEERING CHEMISTRY
= f tan2 4
(13)
= ApenQfd
tan' #J
(14)
Of the three terms on the right of Equation 15, the first is obtained from Figure 4 (at Q / Q f d = 0) and the second from Figure 2 by using, in place off, the value of J i n e f f calculated from Equation 13. The third is calculated from Equation 14. The ratio of ineffective power to the effective (pumping passage) power, Zt, is about 0.2 to 0.3, when calculated at free discharge ( Q / Q j d = I), for the usual screw viscosity pump. All power not used as hydraulic power (QAP) or released as external heat from the pump must heat the fluid in the pump. As a result, the viscosity of the fluid will decrease. with corresponding decreases in po\\rer and performance. If such heating is extreme, the power
a" a \
a
a
0.00
0.20
0.60
0.40
Q Q fd
0.80
1.0 0
EXTRUDER FOR PSEUDOPLASTIC LIQUID
0
.2
.4
Q -
.6
.8
1.0
Qfd Figure 5. Theoretical power efficiency is useful as guide to design, most efficient use of power being at Q / Q f d from 0.3 to 0.8
is a zone at the entrance to the screw where the groove has been deepened. I n rating the pressure gain of the pump, of course the suction head requirement, being a pressure drop, must be deducted, along with all other internal pressure drops, from the calculated pressure gain.
. . quirement and performance of the pump can be estimated by dividing the pump into a suitable number of small sections for analysis, calculating the temperature rise in each section, and correcting the viscosity from the temperature rise. Design Corrections. Because the equations thus far presented are based implicitly on infinitely broad plates, a wall correction must be employed for design. Rowel1 and Finlayson (8) recommend a correction factor of [l 0.63 ( h / w ) ] to be applied to Q found above. The correction factor is adequate up to h/w = 0.5. Although the correction is strictly applicable only to Newtonian fluids, it is believed accurate enough for design with pseudoplastic fluids, After this connection is applied to the flow rate from the pumping passage, a deduction must be made for “underflow,” material passing under the land of the threads circumferentially in a screw viscosity pump. In a manner of speaking, underflow is the material which never gets pumped because it adheres to the casing wall in a screw viscosity pump. I n the vast majority of cases the deduction for underflow may be taken simply as ‘/z w*(D h)N6; however, problems involving high pressure or wide clearances may require special treatment involving extending the curves to values of Q / Q f d greater than unity. I n such cases the land is treated as a viscosity
-
+
Experimental
pump operating with a negative pressure gain. Finally, leakage, external and internal, must be deducted from the pump delivery. Internal leakage will be axially across the land of threads in a screw viscosity pump. Equation 8 is usually applicable for estimating both external and internal leakage. However, the effect of imposed motion may significantly reduce the effective viscosity in the direction of the motion. T h e resulting leakage may then be estimated by solving for r in Equation 1 and then using po/(l C T ~in ) place of po in Equation 8. This is especially applicable to bushing seals when the rate of shear between the shaft surface and bushing surface is such that the apparent viscosity reduction, circumferentially, is by a factor of greater than T. I n this event the leakage will be by a spiral flow path, rather than equivalent to simple viscous flow through an annulus. The final calculation is that of the suction head requirement. The fluid must flow in an axial direction into the annular space between the root of the screw and the wall of the casing for a distance approximately equal to the lead of the screw before the pump action can begin. If calculations indicate that the fluid is not under adequate pressure to feed this groove, a “gathering section” may be added. The gathering section
+
Some test data, obtained during preinstallation performance evaluations of four viscosity pumps operating with an experimental polymer solution, are plotted in Figure 7. Q is the observed delivery less the calculated underflow, AP is the observed pressure gain, and Qfd and AP, were calculated from test conditions and pump geometry. Brookfield viscosities were used for po. Two of the viscosity pumps were roll pumps, which consist of a revolving cylinder in a partially annular case and in which the inlet is separated from the discharge by a barrier, or scraper blade. The roll pumps supplied the higher shear data ( 3 ) . The other two viscosity pumps were single-screw extruders. The pseudoplasticity constant, C, was evaluated as 2.4 X ft.4/lb.12 for the experimental polymer solution using Equation 2 and pressure drop measurements. Because the data were taken originally for mechanical evaluation, rather than for support of a design procedure, many uncontrolled errors are responsible for the scatter of the data. Nevertheless, the high shear data tend near the f = m curve, and this fact is considered significant. The data are of inadequate precision and shear range in the low delivery region, however, to lend support to the design method in this region. A more detailed analysis of the data is available ( 3 ) . VOL. 51, NO. 7
JULY 1959
815
GEOMETRY
WIDTH n
U
LENGTH PER TURN
I
Q f,i Figure 7. Pump test data, although scattered because of poor data precision, partially confirm the design analyses
Figure 6. Transformations are necessary to express single screw extruder geometry in terms of idealized parallel plates
Conclusions
The design methods outlined represent a rational, rapid, and accurate approach to the design of a screw viscosity pump. From pressure drop measurements through short lengths of tubing alone, it is possible to proceed directly to the design of the desired screw viscosity pump, using the methods outlined. The primary limitation of the method is the goodnes of f i t of the rheological equation over the range of shear rates encountered for the fluid to be employed. Nevertheless, the design method is felt superior in principle to a method of utilization of a superficial “average” viscosity reduction, even when only fair fits of the rheological equation are possible. Acknowledgment
T h e advice and encouragement of C. F. Oldershaw were very helpful during the development of the analysis and the writing of this article. W. J. Backer contributed many of the data. F. V. Lyle arranged the machine computation for preparing the design charts. The cooperation of The Dow Chemical Co. in giving prrmission to publish this article is gratefully acknowledged. Nomenclature
b
= thread (groove) width, meas-
C
=
D
=
ured axially, feet constant of pseudoplasticity, ft.4,’lb. force2 inside diameter of tube, or screw mean-Le., pitch-diameter, feet land width, measured axially, feet shear rate index, defined as
C
(y)*,
dimensionless
h
= distan‘ce between surfaces, feet
k
=
arbitrary proportionality constant L = axial length of flighted section of screw, feet; (with subscripts) length in direction of flow, feet = passage length, feet L, = passage length per turn of LIC groove, feet = exponent, dimensionless n N = rotational speed, revolutions per second AP = pressure difference, lb. force/ sq. ft. (gain for pumps, loss for tube) rate or delivery, cu. ft. per Q = flowsecond t = lead of screw, feet = surface speed, feet per second V = local fluid velocity (at distance u y), feet per second = channel width. feet W wineIf = width of channel for ineffective flow component, per turn of groove: feet = distance normal to direction of Y u, feet, measured from moving surface 2 = power, ft.-lb. force/sec. 8 = clearance (radial) between lands and casing, feet 4 = mean helix angle, defined as t arctan -, radians
TD
p 7-
= viscosity, Ib. force-sec./sq. ft. = shear stress, Ib. force/sq. ft.
SUBSCRIPTS
= channel, or passage
;d
ineff
e
=
f
=
8 16
INDUSTRIAL AND ENGINEERING CHEMISTRY
a t free discharge-Le., a t zero pressure gain = ineffective-i.e., pertaining to flow normal to mean helix angle
=
m
= a t moving plane
0
=
S
= at shutoff-Le.,
sn
t
a t zero shear stress a t zero net flow = at shutoff for a Newtonian fluid with viscositv equal to the zero-shear viscosity of the pseudoplastic fluid = theoretical
Any consistent set of units may be used. literature Cited (1) Alves, G. E., Boucher, D. F., Pigford, R. L., Chem. Eng. Progr. 48, 385--93 (1952). (2f Carley, J. F., othcrs, IND.ENG.CHEM. 45, 969-93 (19531. (3) DeHaven, E. S., Chemical Engineering Conference, Montreal, P.Q., Canada, April 23, 1958. (4) Eccher, s., Valentinotti, A., IND. END.CHEM.50, 829-36 (1958). (5) Jackson, M. L., Laracot, F. J., Richards, H. R., Ibtd.,. 50,. 1569-76 (1958). (6) Mohr, W. D., Sauton, R. L., Jepson, C. H., Ibid., 49, 1857-62 (1957). (7) Mooney, M., J . Rhrol. 2, 211-13 (1931). (8) Rowell, H. S., Finlayson, D., Engineering 126,249-50, 385-7 (1928). (9) Wiley, R. M., Pierce, J. E., Chem. Eng. Progr. 47, 432-5 (1951).
RECEIVED for review June 18, 1958 ACCEPTED April 13, 1959 Material supplementary to this article has been deposited as Document No. 5906 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 25, D. C. A copy may be secured by citing the document number and by remitting $6.25 for photoprints or $2.50 f o r . 35-mm. microfilm. Advance payment IS rquired. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
