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ROY FORESTI, Jr.,l and TUNG LIU Monsanto Chemical Co., Dayton 7, Ohio
How to Measure Power Requirements f o r .
..
Agitation of Non-Newtonian Liquids In the Laminar Region A new method is developed for estimating laminar-region power consumption from liquid properties, agitator speeds and gross dimensions, and vessel size. It applies to Newtonian and pseudoplastic liquids, and holds for several varieties of agitators
THE
primary objective in the design of a stirred agitator is to obtain a specified degree of mixing in a liquid of known physical properties. Unfortunately, ”degree of mixing” is difficult to define and even more difficult to measure; some secondary variable such as power consumption for a given speed must be used. Theoretically, for a rotating disk in an infinite body of ideal viscous fluid, power consumption at low speeds can be calculated (2); for complex agitator geometries, this becomes impcrssible. Currently, power consumption is related to agitator speed and liquid properties through dimensionless groups and empirical correlations. These correlations suffer from two shortcomings : They refer to specific agitator types, and do not permit calculation of power requirements from rheological characterization of the liquid being agitated. A generalized correlation for the laminar region corrects these shortcomings. This work is part of a continuing study of agitation phenomena,
which will include studies in the turbulent region of stirred vessel performance. Agitation System T h e vessel used for agitation power measurements was machined from steel pipe and has a capacity of approximately 6 gallons. Four baffles, each inch wide, can be easily inserted or removed to test their effect on agitator power requirements. A drill press with
Literature Background Prediction of power consumption in agitation of non-Newtonian fluids Mixing of high viscosity Newtonian and non-Newtonian liquids Practical mixing technology Agitation of non-Newtonian fluids Use of pilot plant mixing data Power characteristics of mixing impellers Heat transfer to viscous material i n jacketed agitated kettles
(4) (6) (7) (10 ) (11) (1%
Present address, Special Projects Division, University of Dayton Research Institute, Dayton 9, Ohio.
860
p/
DRILL PRESS ARBOR
Agitation power requirements are measured in a vessel machined from steel pipe
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INDUSTRIAL AND ENGINEERING CHEMISTRY
modified pulley system drives the agitators. A l/$-hp. variable-speed direct current motor provides continuous variation of agitator speed from 0 to 2000 r.p.m. The torque developed by this motor is detected by a strain gage system and indicated by a potentiometer type of recorder. All measurements were made at room temperatures. Four agitators were studied: a flatbladed turbine, an anchor: and t\co sizes of cones (drawn to scale. in Figure
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1). Theoretical as well as practical considerations governed the choice of agitators. Unlike the turbine and anchor, the cone agitator is not generally used industrially and has been given only passing mention in the literature. As there was no past experience to draw upon in selecting cone dimensions, two agitators were used, with dimensions chosen to “straddle” values that would be anticipated as optimum. The individual cone sections were mounted to permit the spacing between the sections to be varied. Measurements were made a t separations of 0, l / q , and In general, the results for cone separation were similar to those for 0-inch separation; thus, reference to “open” cones denotes l1/2 inch separation and “closed” cones denotes no separation. T h e cone agitator possesses smooth surfaces, parallel to the liquid streamlines. Instead of generating a rotating wake as do the turbine and anchor, the cone produces a rotating boundary layer. The liquid flows along the inner andouter surfaces to the point of greatest diameter and then radially away from the cones. Thus, the bulk flow of liquid is like the pumping action produced by the turbine. Test liquids
6“
Liquids used were a Dow-Corning 200 silicone fluid, a 10% solution of sodium carboxymethylcellulose (CMC) in water, a 570 solution of polyisobutylene in Decalin, and a 7770 suspension of Catalpo clay in water. A concentric cylinder viscometer, made in this laboratory, was used to measure the flow properties of each liquid. I t had the orthodox cup and bob geometry and plotted the bob torque directly and continuously as a function of cup speed. Cup and bob surface temperatures were measured by thermistors. The flow characteristics liquids are shown in Figure shear rates were calculated by the method of Krieger and Elrod (4)and the points shown on the curves were calculated from the viscometer plots. The silicone was used as a standard or “control” liquid because of its Newtonian flow properties and low viscositytemperature coefficient. Large temperature gradients are generated during the agitation of viscous liquids because of poor bulk mixing and conversion of large quantities of localized mechanical energy into thermal energy. I n liquids having high viscosity-temperature coefficients, viscosity varies widely throughout the vessel. The silicone liquid also has physical and chemical stability. Catalpo clay (Southern Clay, Inc.) consists of approximately 37% AlzOa, 44% SiOz, and the remainder organic
CONES
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Figure 2. at 23’ C.
Flow characteristics of the test liquids
VOL. 51, NO. 7
JULY 1959
861
matter; particle size is 78% below 2 microns. Water suspensions were stable. For non-Newtonian liquids the relationship between shear stress and shear rate can be complex. A fair approximation is the power-type function 7
=
B
($)n
where r = shear stress, du/dy = shear rate, and B and n = constants characteristic of the liquid. For a Newtonian liquid n = 1 and B is the viscosity. This model, used by numerous investigators, has mathematical simplicity and a minimum of defining parameters. In Figure 3 are shown plots of log r us. log du/dy for the test liquids. The data are not fitted by straight lines, except for the silicone material, which is Newtonian. Models other than that defined by Equation 1 gave a better fit of the experimental data, but when they were used to evolve a dimensionless quantity for fluid flow systems, the results were unobtainable or in an impractical form. Consequently, average
1
2
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IO2 SHEAR RATE sec:l
Figure 3 .
Log-log plot of test liquid characteristics
Data are not perfectly fltted b y straight lines
10,ooc
IO00
I I .
100
u)
E/%
BL
H
IC
I.( 0
0.I
Figure 4.
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1.0
10
Power number-Reynolds number correlation is best for Newtonian liquids
INDUSTRIAL AND ENGINEERING CHEMISTRY
100
NON-NEWTONIAN FLUIDS straight lines were drawn through the points on Figure 3 and B and n were calculated (Table I).
10,000
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CONE 4” CLOSED
Table 1. Viscous Properties of Test Liquids at 23” C. W e r e Calculated B , DyneSec.n/Sq. %(DimenCm. sionless)
D.C. 200 silicone 10% CMC 5 % polyisobutylene Catalpo clay
298 521 816 19 0
1.00 0.52 0.34 1.63
Previous investigators (6, 12, 13) have expressed agitation power results for Newtonian liquids in the laminar region by the dimensionless equation
where P = agitation power, p = liquid density, d = agitator diameter, N = agitator speed, p = liquid viscosity, and K = constant. A consistent set of dimensions is used. ( d z N p ) / p i s the commonly used Reynolds number for is a dimenthe agitator and Pg/(pd5N3) sionless group called the power number. For a non-Newtonian liquid which follows a power-type function represented by Equation 1, the above Reynolds number is of the form (3, 8) : (3)
As the Reynolds number defined by Equation 3 is concerned only with the agitator and the liquid, and in no way affected by the vessel, each agitator is correlated by a different value of K in Equation 2. A more desirable Reynolds number would describe the entire system-agitator plus vessel-and eliminate the customary family of curves. A semiempirical development along these lines resulted in a modified Reynolds number:
where H = depth of liquid, h = height of agitator, and D = diameter of vessel. The effectiveness of this type of correlation is demonstrated in Figures 4 and 5. The data in Figure 4 were obtained from the four agitators, the four test liquids, and both baffled and unbaffled conditions. The correlation is best for Newtonian liquids, and as n decreases the deviation increases. For n > 1, no correlation was obtained. The lines shown on Figure 4 represent average values determined by regression analysis on a digital computer. The values for K’ in the equation
p’8d a ~=~K’ (Re’)-’
(5)
I 0004
001
01
10
10 0
30
*(+;(-&,
Figure 5. The correlation for any one agitator is better than the combined correlation for all of the agitators
are presented in Table 11. The data in Table I1 apply only to modified Reynolds numbers below 4.0, where flow is strictly laminar.
Table 11.
an agitation system should apply to a range of liquid properties as well as agitator types. Such a correlation was obtained which included all the agitators and liquids (except the dilatant material) used in this study.
Experimental K’ Values
for Equation 5 K’ Range (95% K’ (Av.) Confidence) D.C. 200 silicone 10% CMC 5% polyisobutylene
160 32 8.6
129-1 98 2345 3 7-22
Unbaffled data are plotted for the individual agitators in Figure 5 (baffled data fall on the same lines, but are omitted for clarity). Also shown are here the conventional (Equation 3) is used for the abscissa. The main virtue of a single line correlation (as in Figure 4) is that it permits the designer to approximate power requirements for an agitated system (vessel, liquid, agitator) for which specific test data are not available. Further refinement of the Reynolds number or power number might produce even less divergence of individual agitator characteristics than is shown in Figure 5. The optimum type of correlation for
where A is an empirical constant which makes the three lines of Figure 4 approach the data for Newtonian liquids. Three materials are insufficient to determine A exactly; an approximate value based on this work is 50. Data for the Catalpo clay could not be correlated in the same manner as the other liquids, because its power number increased directly with speed. A portion of the data for this material (for 4inch cones) is included in Figure 4 for comparison with the other liquids; complete data are presented in Figure 6. Dilatant-type liquids are not correlatable in the same manner as pseudoplastic liquids. This is not serious, however, as dilatant-type liquids are relatively rare (in fact, it is difficult to find a dilatant liquid that flows easily-Le., more fluid than the classical example of wet beach sand). Even though the Catalpo clay suspension used in these studies was 77y0by weight, the material appeared far less viscous than any of the other liquids. VOL. 51.
NO. 7
JULY 1959
863
Figure 6. Effect of stirrer on power number- Reynolds number correlation for Catalpo clay suspension Results of agitation of the clay suspension were graphically different from those of pseudoplastic liquids
liquids. Mihen agitating Xehvtonian or pseudoplastic liquids, the baffles had no effect on the power requirements. 1n contrast are the results from tests on the dilatant liquid. Here (Figure 6) the baffled condition required more power than the unbaffled. This is undoubtedly due to the shift of liquid movement away from the agitator. The baffles disturb the flow under these conditions more than in the case of the other types of liquids and as a result increase the power requirements. Acknowledgment
The authors express gratitude to H. R. DuFour, J. S. Stanton, Eric Barnes, and S. B. McKee for assistance in conANCHOR
structing equipment and recording data, to J. R. Fair for assisting in preparation of the manuscript, and to the Monsanto computer group for assisting in regression calculations.
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The range of Reynolds numbers for the clay suspension is unfortunately narrow and limited to the transition region, because of the properties of the suspension. At very low agitator speeds (corresponding to low shear rates) the shear stress was much lower than for any of the other Lquids tested (Figures 2 and 3). At the higher agitator speeds (corresponding to high shear rates) however, the shear stress became larger than the other liquids. Thus. the range of speeds (and hence range of Reynolds numbers) over which torques can be measured accurately is very small ‘The Reynolds number variation is further shortened because the exponent of N ( = 2 - n = 0.37) is so small.
Flow Patterns i n Vessels The designer should recognize that such correlations may not completely define the fluid dynamic patterns within the vessel. When the polyisobutylene and CMC solutions were agitated at the lower speeds (by any agitator but the anchor). the surface became hemispherical, convex upward. At these conditions, the pumping pattern was almost the reverse of that in a Newtonian liquid. As the speed of agitation was increased, the liquid surface became level again and finally developed the normal type of vortex. The transition from hemisphere to vortex always occurred at approximately 1200 r.p.m., regardless of the agitator used. The effect of elastic properties has been described for rotating cups (70) (Weissenberg effect). For some agitation operations this effect could be serious. Theoretical considerations, confirmed
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experimentally by Metznet and Taylor (Q), show that the shear rate decline between agitator and vessel wall is much greater for pseudoplastic than for Xewtonian liquids. Thus, the effect of vessel wall (and baffles) on the agitator power is decreased; these studies showed no power increase resulting from the presence of baffles. Results from agitation of the clay suspension were graphically different in all respects from those of the pseudoplastic liquids. When this suspension was stirred slowly, it appeared far less viscous than the other liquids. Visually, the agitation was very similar to that of the Kewtonian material-normal motion, vortexing, etc.-but the measurements told another story. As the agitator speed was increased, the torque required increased at an exceptionally high rate, to produce the peculiar results shown in Figure 6. Even though this material appeared very fluid, moderate shear rates caused it to solidify (deposits formed after prolonged agitation). As the viscous forces become very large at increased shear rates, the fluid near the agitator produces the same net result as if the diameter of the impeller were increased. As power number is very sensitive to impeller diameter ( P -l/&). this overshadows speed effects. Thus, power number was calculated by use of the impeller diameter, but the impeller shaft was reacting to some higher value of diameter. This is equivalent to saying that for dilatant liquids, the region of most liquid movement-i.e., volume per unit of time-is shifted away from the agitator. For pseudoplastic liquids, it is shifted toward the agitator. The effect of baffles demonstrates the difference in flow conditions for pseudoplastic liquids, compared with dilatant
INDUSTRIAL AND ENGINEERING CHEMISTRY
B
= viscous property of liquid as
defined by Equation 1 vessel diameter H = height of liquid in vessel .Y = speed of agitator P = agitator power d = diameter of agitator 4 = dimensional conversion factor h = height of dgitator n = viscous property of liquid as defined by Equation 1 dutdjl = shear rate ,u = liquid viscosity p = liquid density 7 = shear stress .4ny consistent set of units mal br used.
D
=
literature Cited
(1) Calderbank, P. H., Moo-Young, M. B., Trans. Inst. Chem. Engrs. (London) 37, 26 (1959). (2) Cochran, B. A,, Proc. Cambridge Phil. Soc. 30, 365 (1934). (3) Duncan, M‘. J., “Physical Similarity and Dimensional Analysis,” p. 7 2 , Edward Arnold & Co., London, 1953. (4) Krieger, I. M., Elrod, H., J . A,b,bl. Pizys. 24, 134 (1953). (5) Lee, R. F., Finch, C. R., Wooledgr, J. D., IND.ENG.CHEW49, 1849 (1957). (6) Lyons, E. J., Chem. Eng. Progr. 44,
341 (1948).
(7) Markovitz, H.: Williamson, K. R., Trans. SOC.Rheol. 1, 25 (1957).
(8) Metzner, A. B., Otto, R. E., A.T.CI1.E. Journal 3, 3 (1957). (9) Metzner, A . B., Taylor, J. S., “Flow Patterns in Agitated Vessels,” A.1.Ch.T:. Annual Meeting, Cincinnati, 1958. (10) Reiner, M., “Deformation and Flow,” H. K. Lewis, London, 1949. (11) Rushton, 3. H., Chem. Eng. Progr. 47, 485 (1951). (12) Rushton, 3. H., Costich, E. W., Everett, H. J . , Ibzd., 46, 395, 467 (1950). (13) Uhl, V. W., Chem. Eng. Progr. Sym$. Ser. 51, No. 17, 93 (1955).
RECEIVED for review January 14, 1959 ACCEPTEDApril 30, 1953
