m 1
'!
a
..n c
CLOSE-CLmRANCE AGITATORS w. noy renney n. J.
oeii
POWER REQUIREMENTS
TRANSFEI COEFFICiENT!
CLOSE-CLEARANCE AGITATORS
Close-clearance equipment is most frequently used for high viscosity fluids in which it is not practical to produce turbulence or where a fouling deposit or a solid precipitating phase crystallizes on the vessel wall. The following is a summary of fluid processing equipment which employs close-clearance agitators. Anchor Agitators (Figure 1). They are used in pot-type heat-exchange or chemical reaction equipment. The horseshoe-shaped anchor has a fixed clearance between its edge and the vessel wall. Anchors are generally used for moderate to high viscosity liquids (10 to 1000 poises). Extruders (Figure 2). Extruders are used for the processing of high polymers and other very viscous materials. The rotating screw both mixes the material and pumps it through the extruder. The clearance between the screw and barrel is usually constant. In normal extruding operations the barrel is frequently electrically heated; however, the energy imparted to the material as a result of viscous shear is considerable. Extruders are frequently used to cool a polymer prior to pelletizing. For this application the viscous dissipation is to be minimized and the heat transfer through the extruder barrel is to be maximized. Thii Film Equipment (Figure 3.) These devices are widely used as evaporators, chemical reactors, etc., for a variety of materials. Their most frequent application is with heat-sensitive materials because of their short residence time. The clearance between the agitator and vessel wall can be either fixed or variable because both rigid and hinged agitators are ped. The processed liquid is continually spread in a thin layer over the vessel wall; it moves through the device either by the action of gravity or of the agitator (a tapered or helical agitator will produce longitudinal forces on the liquid) or by both.
PART
POWER REQUIREMENTS
~
W. Roy Penney K. J. Bell
he technical literature is increasing at such a fast rate that it is difficalt to keep abreast of significant developments in even a limited area without some help in condensation and interpretation. The intent of this paper is to present a summary and critical review of literature pertaining to heat-transfer and power requirements for liquid processing equipment employing closeclearance agitators.
T
EQUIPMENT EMPLOYING CLOSECLEARANCE AGITATORS I t is first necessary to define how we are using the term “close-clearance.” “Close-clearance” is used to describe the class of fluid processing equipment which employs agitators (anchors, scrapers, helical ribbons, extruders, etc.) that sweep practically the entire vessel volume. Then, not only is the clearance between the agitator and vesscl “small,” but also the agitator length is approximately equal to the vessel length. This designation is necessary to distinguish this equipment from those which employ agitators (propellers, turbines, paddles, etc.) that sweep only a small fraction of the vessel volume. For this latter class the clearance between agitator and the vessel wall is generally large, and the agitator length is small compared to the vessel length. Much of the equipment which we call “dose-clearance” has heretofore been called “scraped-surface” equipment. This term is really a misnomer because, almost invariably, a film of liquid will exist between the agitator and vessel wall. “Scraped-surface” will only be used here to indicate that the vessel wall is scraped clean. Close-clearance equipment employs a myriad of agitator configurations. From a predictive standpoint, it is convenient to classify these configurations as either lked-clearance or variahle-clearance. Fixed-clearance equipment employs rigid agitators. In variable-clearance equipment the agitators are forced toward the vessel wall hy springs, centrifugal action, hydrodynamic action of the fluid on the agitator, and/or by the fluid friction between the agitator tip and the vessel wall. The “slipper bearing effect” acts to hold the agitator off the wall; in general, the clearance varies with operating conditions. 40
INDUSTRIAL AND ENGINEERING CHEMISTRY
E-
Figura 1. The amhr agitator
Figwe 2. Doubl&jighled
SCICU)
urnrdar
Votator (Figure 4). The votator (a trade-mark of the Chemetron Corp.) has floating scraper-agitators (two or more) which are forced against the cylinder wall by the hydrodynamic action of the fluid on the agitator and by centrifugal action. The blades are loosely attached tr, a central shaft called the mutator. The Votator is used extensively in the food processing industry and in the manufacture of greases and detergents. As the blades are free to move, the clearance between the blades and the wall will vary with operating conditions. The Votator is normally 3 to 4 inches in diameter and 2 to 4 feet long. Spring-Loaded Scrapers (Figure 5 ) . This class of equipment has scrapers which are usually held against the wall by leaf springs. Typical applications are processing of heavy waxes, heavy oils, and crystallizing solutions. These scrapers will also have variable clearance between agitator and cylinder wall because the spring force will be balanced by the radial hydrodynamic force of the liquid on the scraper. The units are generally 6 to 12 inches in diameter and up to 40 feet long. Helical Ribbons, Augers, a n d Twisted Tapes. Some of the more specialized heat exchangers and chemical reactors employ helical ribbons, augers, or twisted tapes as agitators. They are generally used for high viscosity materials. They are fixed-clearance devices. There is no general rule as to maximum or minimum dimensions since each application is a special case. Equipment for Extremely High Viscosity Materials. Materials with viscosities above 100,000 cp. generally require more specialized equipment for efficient fluid processing. Typical equipment includes kneaders, roll or pug mills, and double-arm mixers. They usually have fixed clearances between agitator and vessel wall.
ments are in order. The work on process agitators (excluding extruders) has been, for the most part, concerned with fitting dimensionless parameters to experimental data. Dimensional analysis is successful insofar as the power data for any particular process agitatorgiven complete geometrical similarity-can be correlated for a Newtonian fluid by plotting the conventional power number (P) us. the rotary Reynolds number (Re). This correlating method, then, accurately reflects the effect of fluid properties. I t will be helpful later to discuss the form of this correlation-for a fixed geometry-as we go from the viscous regime (inertia effects are negligible) through the transition regime and into the turbulent regime. Generally, below a Reynolds number of 20, the curve of P us. Re has a slope of -1 on log-log paper. This regime is called “viscous” because inertia forces are negligible. I n the range of 20 < R e < 10,000, the slope gradually changes from -1 to 0. This is the transition regime where both viscous and inertial effects are important. We must emphasize that the flow can be laminar in this regime. I t appears from Uhl and Voznick’s (77) heat-transfer data that turbulence probably starts about Re = 600. Then in the range 20 < R e < 600, the flow is laminar. Inertial effects become increasingly important as R e increases; in the range 600 < R e < 10,000, the flow becomes increasingly turbulent as Re increases. Above Re = 10,000 the curve of P us. IC
D
Figure 3.
eva*orator
REVIEW A N D ANALYSIS OF THE LITERATURE The justification for reviewing all the widely different devices just discussed under the single classification of close-clearance equipment is the need to generalize design correlations. For example, we will show that the correlating parameters for an anchor agitator can be induced from previous theoretical work on screw extruders. I t should even be possible to use-in conjunc‘tion with methods for predicting clearance-correlations developed from fixed-clearance devices for variableclearance equipment. The common thread that binds all these devices into a single class is really not the close clearance, but rather it is the fact that close-clearance agitators almost always are as long as the containing vessel. However, we are not proposing that a completely general correlation can be developed; on the contrary, with the present state of the art, it is necessary to limit a single correlation to a single type of agitator. The important point is that the same general correlating method will suffice for all close-clearance devices. Before reviewing specific articles, some general com-
Figure 4.
Figure 5.
The Votator
scra~ers
VOL. 5 9
NO. 4
APRIL 1967
41
Re is essentially flat because the power becomes independent of viscosity (and therefore Re) in the completely turbulent regime. Close-clearance agitators are most frequently used with viscous materials. For this reason most of the power data is in the viscous regime (Re < 20) with a considerable portion extending into the transition regime (20 < Re < 10,000). Table I presents a summary of the experimental investigations of power requirements. Dimensional analysis adequately correlates the effect of fluid properties, but geometrical variables have defied correlation by this technique. Extruders. As contrasted to the work on process agitators, the work on extruders has been almost totally of a theoretical nature. This work provides a sound basis (within the limitations of the assumptions) for the induction of correlating methods for other equipment. Almost all analyses have been directed toward predicting power requirements by solving the Navier-Stokes equations with inertia terms neglected. U p to the mid1950's this work is well summarized by Squires ( I ) . More recent work is covered by Squires (75) and Booy (2). It is rather surprising that no previous investigator of ribbons and augers has mentioned this important body of work. I t is beyond the scope of this paper to consider the solutions in detail. Our primary interest is to ascertain the parameters which will suffice to correlate power for other close-clearance agitators. Those interested in power requirements of extruders are referred to the original works. The power requirements of the extruder screw are assumed to be the sum of those for each of three processes, which are separately analyzed :
1. Power consumed between the flight edge and the extruder barrel (called clearance ejects hereafter in this paper) 2. Power dissipated by viscous shear in the screw channel (called bulk eJects hereafter) 3. Power required to raise the fluid pressure Only clearance and bulk effects will be considered here since the third effect is usually not present in other close-clearance equipment and is easily computed from an energy balance. The clearance effects are calculated assuming the clearance is so small that the barrel and flight edge approximate two flat plates. By straightforward analysis the following expression for power is obtained for a two-blade agitator :
):(
@, = 2 ad2N2
(k)
tors did ) and that the clearance between the screw and the barrel is small. Mohr and Mallouk ( I ) have obtained a solution which takes into account the effect of the clearance on the channel power. Although all these solutions are approximate, they do give the pertinent dimensionless parameters and suggest the correct functional form of correlation. These solutions result in the following relationship for bulk power :
p,
=
):(
4 n3dZN2
[ f ( C / d , d,/d, h/d)]L
We are primarily interested here in how a variation in C affects power because many of the agitators of interest are straight with d, = 0 ; consequently d,/d and lz/d will not enter the correlation. We note that for both clearance and bulk effect that an increase in C decreases the power. Helical Ribbons and Augers. Bourne and Butler ( 3 ) have taken data for one double-flighted helical ribbon with two non-Newtonian fluids, for which n = 0.5. They used the analytical solution for a power-law fluid for the power consumption between two concentric rotating cylinders to ascertain the parameters
T A B L E I. E X P E R I M E N T A L Test
Investigator and Reference
Fluids
Nagata et al. ( I 7)
Water and millet jelly solution
Foresti and Liu (7)
Silicone fluid; 10% C M C in water; 5% polyisobutylene in decalin; 77Yc catalpoclay in water
Uhl and Voznick (17)
Pennsylvania cylinder oil
Skelland and Leung (74)
Aqueous glycerol
Calderbank and Moo-Young ( 5 )
Fuel oil, treacle, glycerol, aqueous glycerol, CMC clay 4-kaolin polymer
Hoogendoorn and den Hartog ( 9 )
Oil, glycerin-water and molasseswater mixtures
Chapman and Holland (6)
Corn sirup-water mixture
L' ~~
Booy (2) has done the most recent work on predicting the bulk-power dissipation. He has considered screw channel curvature, whereas previous investigators had approximated the curved channel with a straight channel. He assumed that the channel depth is small compared to the screw diameter (as previous investiga42
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Bourne and Butler ( 3 )
Two CMC-water mixtures and two Celacol-water mixtures
Gray (8)
Corn sirup-water mixture
which would correlate the non-Newtonian behavior. The result is
helical ribbons and augers in a draft tube. Their prime interest was mixing. Power data for one of the ribbons were reported to be represented by
(3)
(P)(Re) = 300 (5) Gray (8) conducted limited tests with double-flighted helical ribbons and with centered and off-center augers. When compared on a plot of log P us. log Re, his helical ribbon data were about 40y0 higher than those of Nagata ( I 7). Hoogendoorn and den Hartog ( 9 ) have taken data for double-flighted helical ribbons and single-flighted augers in draft tubes. All agitators of a given type were geometrically similar; therefore, a single curve of log P us. log Re correlated the data for each agitator type. They were primarily interested in mixing. Chapman and Holland (6) have taken data for both centered and off-center augers in cylindrical vessels with and without baffles. They fitted the data in the viscous regime (0.02 < R e < 20) by use of:
pa
= r2- d2N [ n ( l gc
- [d/DI2'")
which may be put in dimensionless form:
where Re' is the Reynolds number for a power-law fluid. They plotted log P/F us. log Re' and obtained a n excellent correlation. We must note that they tested a single agitator ; therefore, they have only correlated the non-Newtonian behavior. Also, n (and therefore F ) was essentially constant ; consequently, the correlation could very well fail if d/D were changed. I n essence they have shown that for a power-law fluid Re' will correlate power for fixed geometry and constant n. T o fully substantiate the concentric cylinder model for other agitators, both n and geometry would have to be varied. Nagata ( 7 7) tested several geometrically similar INVESTIGATIONS OF POWER REQUIREMENTS Vessel Agitators Dimensaons Emfiloyed d Anchor or paddle D = 10, 20, 30 cm. Double-flighted helical ribbon
D
Single-flighted auger in draft tube
D = 10 cm. D' = 5, 7 cm.
Anchor
D = 1 1 . 5 in.
= 10,
20, 30 cm.
260
D
= 10.25, 23.5 in.
- log
Agitator Dimensions = L = 8, 16, 24 cm.
Re)}
{(d,D)
(2.18 ;,og
Re
Comments They were primarily interested in mixing.
d = L = h = . 9 . 4 , 19, 28.5 cm.
They measured power data but did not re. port it. Correlation for one ribbon given in form of P us. R e
d = 4.5, 6 . 3 , 6.51 cm.
L = 9 cm.; h = 3, 4.5, 8 . 7 cm. d = 11.125in.
L = 51/2 in. Anchors
2.38
Use height of liquid in correlation; however, do not report height of liquid n = 0.34,0.52, 1 . 0 , 1 . 6
d = 10.156, 9.75, 8.75, 23, 22.5, 22,
21.5 in. h = 53/16, 23'/2 in. Votator
D
= 3.0in.
Anchors
D
= 7 in.
Double-flighted, helical ribbon
Used 3, 4, 5 floating blades on mutator of 1.8-in. diam. and 2 blades on mutator of 1-, 1.4-, 1 ,8-, 2.25-in. diameter L = 18 in.
Viscosity varied only slightly. Neither axial flow rate nor mutator diameter affected power. Data reported
L = 3.7; d
Data not given. Cannot reduce dimensionless correlations to other forms
= 5.91, 6 . 3 , 6 . 5 3 i n .
D
= 10.5 in.
D
= 15 in.
L L L L
D
= 24, 180 cm.
d = 23, 173 cm.
L D = 24, 180 cm.
Single-flighted auger in draft tube
D' = 11, 82.8 cm.
Single-flighted auger
D
= 5.5, 7 .O, 9.5, 1 1 . 5 in.
=
6 . 6 ; d = 5.91, 6.53 in.
= 3 . 7 ; d = 5.91, 6.3, 6 . 5 3 i n .
6 . 6 ; d = 5.91, 6 . 5 3 in. = 6 . 6 ; d = 5.91, 0 . 5 3 in.
=
= 25.8, 189 cm.; h = 14, 105.5 cm.
d = 9.6, 72 cm. L = 30, 225 cm.; h = 11, 82.8 cm. d = 3.0, 5 . 0 i n .
L
= 13.0, 15.0 in.
h
=
Both centered and off-center augers
1.8, 2 . 0 , 4 . 0 in.
Double-flighted helical ribbon
D
Double-flighted helical ribbon
D
= 9 . 0 in.
d = 8.55 in. L = 8.0in.
Single-flighted auger
D
= 9 . 0 in.
d = 3.0 in. L = 9 . 0 in.
= 25.4 cm.
Article in Dutch. We are not sure if all fluids were used with all geometries. No data or data points on curves given
d = 19 cm.
n
0 . 5 for all liquids
L = 31.2 cm.; h = 10 cm. Interested in mixing.
VOL. 5 9
Reports data
NO. 4
APRIL 1967
43
This relationship appears to be incorrect in a t least the following respects:
1. For clearances approaching zero ( d / D -+ 1 . 0 ) ) the power becomes independent of C, whereas the power should increase nearly linearly with C (as C -+ 0, edge effects predominate) 2. For constant d / h and d / D (and thus c l d ) , the dependence of p on auger diameter is@a: d3.2from Equation 6. Compound to p a: d 2 from Equation 2 3. As ( d / h ) --t 0 , p --t 0 Anchor Agitators. Nagata ( 7 7) tested paddles with close clearances and lengths equal to the vessel length (equivalent to an anchor with wide arms) ; however, they do not report power data. Foresti and Liu (7) conducted tests in the viscous regime with an anchor, a turbine, and two-cone agitators using one Newtonian and three non-Newtonian liquids (n = 0.34, 0.52, and 1 . 6 ) . The data were plotted as Pus. R e ' (H/L)"[d/'(d D ) ] on log-log paper. For n < 1 this correlation brought the data of all agitators together for a particular n; however, a separate curve resulted for each n with the curve for n = 1 .0 being 20 times higher than for n = 0.34. For n = 1.6 separate curves resulted for each agitator; their slopes were in the range +8 to +15. For n =: 1 the power can be expressed as
+
p
= 340
(d)
d3N2(L/H)(1
+ C/d)
This equation is deficient in a t least three respects.
1. p a d3 rather than d 2 2. p is essentially independent of Cas C+ 0 p probably should increase slightly as 3. p 0; 1,"; H increases for H > L
Also it is difficult to accept that (H/L)" could in fact correlate non-Newtonian behavior. N o mention is made that H was varied; and, if H were not varied, the correlation could very well be fortuitous. Uhl and Voznick (77) have conducted tests with anchors of two different diameters, each with four different clearances. They present the data as log P us. log Re and a separate curve results for each clearance. The curves for the two different diameters do not merge. They suggest that if P is divided by the ratio of effective peripheral length to vessel diameter, the resulting power number will agree for different anchor diameters at constant C / D . They tested both two- and three-inch wide arms; the effect of arm width on power was negligible. Their data agree within 30YG of the Newtonian, anchor data of Foresti and Liu (7). Calderbank and Moo-Young (5) have used data of Uhl and Voznick (77) and Foresti and Liu (7) with their own data to obtain a general correlation (including non-Newtonian behavior and geometry variables) for anchors. By analogy to pipe flow they induce that the non-Newtonian Reynolds number for agitators should l)]". All geometrical factors be Re' 8l-"[4 n / ( 3 n are included in the power number. Their correlation may be shown to be inconsistent with physical reality;
+
44
INDUSTRIAL A N D ENGINEERING CHEMISTRY
only the Newtonian case will be considered here. After curve fitting and some algebraic manipulation, the following functional relationship is obtained for power (to simplify the expression with negligible loss of generality, w7e assume that w , d and constant C / d ) rather than d2
1/L (for 8L
p+o,c-+o
>> d ) rather than L
T h e Votator. Skelland and Leung (74) have conducted the only investigation bvith the Votator. They investigated the effect of rotational speed and number of blades and presented the data for t ~ 7 oplades as log P us. log Re. Power varied as the number of blades to the 0.56 power. Their tests were in the lower transition regime (90 < Re < 220) ; even so, the slope of log P us. log Re was -1.25. Tl'hen we assume constant viscosity, there is only one possibility for the slope to be less than - 1-namely, the clearance must increase as speed increases. Most variable clearance equipment will exhibit this behavior. Centrifugally Loaded Agitators. Kern and Karakas (70) have used slipper bearing theory to compute the clearance power requirements of a centrifugally loaded, beveled-edge agitator. They arrive at the following expression for power which is supposed to hold as 0 approaches zero :
p,
=
16 .?r3mD2N3 tan 0
(9)
This result is incorrect because the lift force due to the pressure increase under the bearing was neglected. The following expression has been rigorously derived from the slipper bearing equations in (72) and (76) :
Equating the lift force on the bearing to the centrifugal force relates m , C, and a:
Double Arm Mixers. There are apparently no published data for this equipment; however, Parker (73) has recommended the following scale-up formula :
Bowen ( 4 ) showed that this equation is inconsistent with the work of Uhl and Voznick (77) with respect to the dependence on diameter, speed, and blade width. Double-arm mixers have much in common with screw extruders : both have small clearances between agitator and vessel wall, both operate in the viscous regime, and both have agitators which sweep practically the entire vessel volume. One would expect that Equations 1 and 2 for extruders would be superior to Equation 12 for scale-up. Equations 1 and 2 predict that for constant t / C and h / d , the power is proportional to d2 and P.
NEW CORRELATING M E T H O D We shall now attempt to incorporate existing results into a consistent picture which in our opinion is superior to earlier correlations for design and scale-up. Previous correlating attempts mostly involved fitting experimental data with dimensionless groups with little or no consideration given to how the final correlation represented physical reality. These correlations illustrate the important point that a dimensionless correlation should never be used without first reducing it to basic parameters so that one can judge if it predicts what one knows to be true. Frequently, one can ascertain the pertinent dimensionless parameters and even the correct form of correlation from solutions of the basic governing differential equations for limiting cases. We shall use the previously discussed solutions for the screw extruder here. Equation 1 gives the power consumed in the clearance. It can be put in the following dimensionless form:
or Equation 2 gives the functional form of the equation for the bulk power. I t can also be put in dimensionless form (replacing 4 7r3 with P, an undetermined constant) :
These functionals contain a new power number which includes d4L rather than d5. It is now obvious why the conventional power number will not correlate power for different diameters for agitators which have lengths equal to the containing vessel length. Equations 14 and 16 for the clearance and bulk power, respectively, can be combined to give a power number based on total power consumed.
For known clearance, p c can be readily computed if the flow in the clearance is laminar; therefore, only bulk power need be correlated. Handling clearance and bulk power separately will probably be necessary in the transition and turbulent regimes because in these regimes Re is not expected to characterize the fluid dynamic conditions in the clearance. A Reynolds number based on blade tip velocity and clearance ( r d N C / p ) can probably be used to characterize the fluid dynamic conditions in the clearance. We cannot expect to apply a rigorous analysis with the data now available; in fact, agitator thicknesses have not been
reported in most studies so only total power can be correlated here. For particular agitators, variables other than those of Equation 17 will be important. I n general, there are insufficient data to consider other variables. I n one case, data show that an additional variable- is not important-Uhl and Voznick (77) have shown that anchor arm width has a negligible effect on bulk power,
RECORRELATION OF PUBLISHED DATA Anchor Agitators. I n Figure 6, we have replotted the curves of Uhl and Voznick (77) using P,, us. R e with C/d as a parameter. I t was not possible to subtract the clearance power from the lotal power because agitator thickness was not given. If their agitators were no more than ‘/4 inch thick, the clearance power would be less than 8% of the total power for the smallest clearance and even less for larger clearances. The present correlating method makes the data for the inch i.d. vessel consistent with those for the 23l/2-inch i.d. vessel. This correlation should give correct scale-up relationships and probably should give reliable absolute values of power drawn by anchor agitators. Calderbank and Moo-Young’s (5) data are not presented because the parameters of Figure 6 cannot be obtained from their paper. Also, Foresti and Liu’s (7) correlation cannot be reduced to the one of Figure 6 because they do not give the liquid height; however, Uhl and Voznick (77) found agreement with Foresti and Liu (although they do not say how liquid height was obtained) within 30% for about the same size agitator. The Votator. The curve of Skelland and Leung (14) for two blades is replotted in Figure 6. This correlation is only expected to hold for the test conditions because the Votator is a variable clearance device with dynamically and hydrodynamically loaded blades. I t is readily apparent from Equations 10 and 11 that the simple relationship of Figure 6 would not be expected to correlate a dynamically loaded agitator. The Votator curve is shown primarily for reference purposes. Augers and Helical Ribbons. For these agitators, the helix angle (or h / d ) is a n important parameter. The diameter of the center shaft and the ribbon width are also additional parameters ; however, their effect is expected to be negligible for the agitators in common use. Most of the published correlations which could be reduced to the parameters of Figure 7, using only the published information, are recorrelated in Figure 7. The correlation is consistent in the respect that power increases as C / d and h / d decrease, which is what one would intuitively expect. This correlating method provides a sound basis for scale-up as long as h / d , d,/d, and w 7 / d remain constant. There are not sufficient data a t present to justify including any of these parameters iri the power number.
CONCLUDING REMARKS We have attempted to critically review the literature with particular emphasis on the limitations of previous VOL. 5 9
NO. 4
APRIL 1967
45
the Votator. Additional theoretical and especially experimental work are needed here. Almost all experimental work has been done in the viscous or transition regime. More data are needed in the transition and turbulent regime. Even the general form of the P,, us. Re relationship is not established in the upper-transition and turbulent regime; an effort to obtain the data to do this is important.
Gd = 0.0093
.
NOMENCLATURE
Ud
0.01 i
=CJc = minimum clearance between agitator and vessd, ft. C, = maximum clearance betwcen agitator and vesscl, ft. d = diameter of agitator, ft. d . = diameter of agitator shaft, ft. D = diameter of containing vessel, ft. g, = conversion constant, 32.2 lb.,ft./lb., q.see. h = pitch of helix on auger or helical ribbon, ft. H = height of liquid in vessel, ft. K = consistency factor for power law fluid L = length of agitator, ft. rn = mass of agitator, Ib., n = exponent for power law fluid N = speed of agitator, rev./sec. p = power, ft.-lb.f/scc. ps = power consumed in the bulk of the fluid, ft.-lb.,/sec. p . = power consumed in the clearance, ft.-lb.//scc. P, = Ps + A P = conventional power number, pg./pNads Psi = P d P N ’ d ‘ L POL = P a d ~ N ~ f l L P~= L fge/~N’d’L Rc = Newtonian mtary Reynolds number, Na%/rr Re’ = non-Newtonian rotary Reynolds number ( P - “ d ‘ p / K ) t = cimmferential thidmw of agitator nearest vessel wall, ft. w. = width of agitator arm,ft. w, = width of helical ribbon, ft. Greek Letters @ = undetermined constant 8 = angle between agitator and a tangent to the vesscl = absolute viscosity, Ib.,/see. ft. p = liquid density, Ib.,/cu. ft. p (1
C
Vd = 0.0465 I
I
...
F i g m 6, Rccowclation of p d l i h d data for anchor agrlatorr votator
Y
\ \
Figrnc 7. Rtcmtlarion of published dota fm singlcj?ightcd augers and doubicj?ighftd ribbons
REFERENCES
dimensionless correlations. Several correlations were, very simply, shown to be inconsistent with physical reality. Previous investigators have relied too strongly on the standard form and techniques of dimensionless correlation with not enough emphasis on basic fluid mechanics. Investigators have attempted to include secondary variables in the power and Reynolds numbers with insufficientdata to correlate these variables. We have argued, primarily from the theoretical work on screw extruders, that the power number for agitators which are nearly as long as the containing vessel should be based on d‘L rather than ds as is the conventional power number. Correlating data with this new power number will result in power scale-up as dp rather than 8. This is certainly the most important original contribution of this paper. Published correlations for anchors, augers, and helical ribbons ,. Geometrical factors have been correlated using P other than length are not correlated because the published data are insufficient to attempt their correlation. Neither previous work nor this work will suffice to correlate power for variable clearance devices such as 46
INDUSTRIAL A N D ENGINEERING CHEMISTRY
. H. “Some Charactsirtics of Helical ImpUsa in V b Th-RdaM.to Racdce,” Vol. IO, Pap”o.9,A.I.Ch.E. Joint Mcclmg, London (1965). ( 4 ) Bowen, R. L., Jc., “Letter to Editqr,” an.Emg. 73, 7 (March 28,1966). ( 5 ) Caldcrbank P. H. Moo-Young M.B., “Ths PoCharactcri.fics of Aghton for the My!& of kewtonian add Non-Newtonian Fu ld i%”, Tram. Iut. Chm. E n p . (Ladon) 39.22 (1961). (6) Chapman, F. S., H?Upnd. F. A,, ‘‘A Study of Turbine and.Helical--Scrrw Agitatom in Liquid Miring,'. T m u . 1 ~ 1C. h . Engrr. (Ladon) 43, T131 (1965). (7) Forcati R. Jr. Liu, T., “How to Mcaavc Power R uimmmfs for A h t i n of N o n - & w h & Liquids in the Laminar Region,” E%. C h . 51 860 llulv 1959).
( 3 ) L B o ~ u :I ~ an%
.
I d . Chem.%p.
13.
8).
.
. s ... .. . (10) KSn, D. Q., Karnkw H. J. "Mechanically Aided Hcaf Trmfa,.’ C h . Eng. P q r . S p p . S“. 55 (29), 141 11959). (11) Nagata, S., Yanapimoto, M., Yokoyama, T., ‘‘A Study of Mixing of Highly VixousUquids” C h . €88. ( J o p m ) 21 (5),278 (1957). (12)
