AGITATION Power Requirements of Turbine Agitators' ARTHUR W. HIXSON AND SIDNEY J. BAUM Columbia University, New York, N. Y .
The available literature on the power requirements of agitating equipment is reviewed and classified. A dynamometer is the most suitable apparatus for measuring power input to agitating equipment, and a turntable dynamometer is described for measuring the torque requirements of a rotating agitator. An equation for the power requirements of a series of dimemionally similar agitators is derived, and the experimental results are correlated with it. The curve obtained for a n y single agitator design is similar in shape to previously obtained fluid flow curves, and indicates the existence of a laminar flow region, a transition zone, a critical region, and a turbulent flow zone for agitating equipment. The limitations o n the use of models for predicting power requirements are indicated by these curves. A standard turbine agitator design is established, and the effects of single variations from this design on power requirements are investigated. An empirical method is developed for predicting the power requirements of other agitators which differ in more than one manner from the standard design. Comparisons are made between predicted horsepower and measured horsepower for plant size equipment; the predicted values compare favorably. O K E R consumption is one of the prime variables considered in determining the optimum design or operating conditions for agitation equipment. With the present state of knowledge of the unit operation, mixing, evaluation of this variable usually involves construction of the equipment required and selection of a driving motor by a n actual load test or by knodedge of past performance. The shortcomings of this method are obvious: the effect of a variation in fluid properties or equipment design can be only roughly predicted; the time lost and expense involved in load tests appreciably increase the cost of the equipment; and finally, the data obtained from these tests cannot be systematically classified for use in predicting the performance of designs other than the specific one for which the data were
P
1 This is t h e fourth and last of a series of aiticles on Agitation. The previous articles aplieared in April (page 478) and Sovember (page 1433), 1041, and in Jgiiuary (page 1201, 1942.
collected. The purpose of the present investigation is to develop a method for predicting t'he power requirements of one important type of mixing apparatus-the turbine agitator.
Previous Work on Power Requirements of Mixing Equipment Three principal types of impeller design-namely, the paddle, the propeller, and the turbine-are used in mixing the phase combinations liquid-solid, liquid-liquid, and liquidgas. The paddle classification includes impellers which have two or more flat blades arranged symmetrically about' a rotating shaft; the propeller agitator consists of a marine propeller of two or more blades attached t o a rotating shaft; and the turbine classification includes impellers which have two or more flat blades attached a t an angle t,o a rotating shaft.. Sumerous modifications of the three main types occur, and frequently two types are used in combination with each other in the same apparatus. Systematic work on the power requirements of paddle agitators has progressed to a greater extent than on any of the others. Badger, Wood, and Whittemore (.2) reported the power required a t different speeds to drive a, paddle agitator in mater. They measured the net power delivered to the motor between the load and no-load conditions a t various speeds, and pointed out the difficulty in obtaining accurate readings with this method because of meter fluctuations. White and his co-workers (14) made a study of the effects of fluid properties and design on the power requirements of paddle agitators. They measured directly the torque required to drive the vertical shaft. From dimensional analysis these authors developed the empirical equation, p =
0.000129~~.~Z~O.14~~T2.8~~0.SfiD1.1~0.3H0.6
(1)
to correlate their results. I n a later modification of the same data, White and Sumerford (16) showed that it was separated into two distinct regions, one for viscous and the other for turbulent flow, and the nature of the flow depended on the value of the modified Reynolds number, NL2y/z. A similar break in the correlation of power data for paddle agitators was noted by Buche (6) who measured the power input with a dynamometer, the details of which are not given. Buche also pointed out a method for employing models to predict the power requirements of large size equipment, and cited the work of R. Hailer who determined the optimum dimensioiis of a paddle agitator for dissolving rock salt with the least expenditure of energy. An investigation of the power requirements of propeller agitators, comparable to those of White and co-workers and of Buche, has not yet been made. The available data on propellers is meager and of doubtful accuracy. Valentine and RIacLean ( I d ) presented a nomographic chart for predicting the power absorbed by a three-blade propeller rotating in v,-ater, based on pumping capacity at 100 per cent slip. 194
February, 1942
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Bissell (3) illustrated in a series of graphs the effects of speed, size, and viscosity on the power requirements of propellers. The effect of stirrer clearance on the power requirements of a propeller agitator is illustrated by the speed-power curves reported by Badger and McCabe (1). The power consumption at various speeds for a series of geometrically similar turbine agitators was measured by Hixson and Wilkens (9). These authors determined the net load to the driving motor, based on the difference between the load reading and the no-load reading a t the same speed. A Reeves pulley drive was used to vary the speed. Power measurements were also made on baffled vessels, and it was found that addition of baffles enormously increased the power requirements a t any given speed. Hixson and Tenney (8) later measured the power consumed by a turbine impeller operating a t comparatively high speeds (200-1000 r. p. m.) in water and sucrose solutions. Their method was also based on the net load to the driving motor, but correction was made for stray power and copper losses. Hixson and Luedeke (7) measured the horizontal component of wall friction for a series of geometrically similar turbine agitators. The fluid properties were varied by employing water and sucrose solutions, and it was found that the power lost to wall friction could be represented by P = ap0.79p0e1n2.79d3.58(d4h) sin (1.130 - 12) (2)
+
for Reynolds numbers (nd2p/p) greater than lo6. There was also some indication of a critical Reynolds number in the correlation of their data. Valentine and MacLean (12) indicated the effects of size and viscosity on the power requirements of turbine agitators, and MacLean and Lyons (10) showed the effect of pumping capacity on the power requirements of turbine agitators. Catalogs of the New England Tank and Tower Company (11) give extensive compilation of speed-power data for turbine agitators under specific conditions. These power data are for plant size installations. The powerseported is net, based on the predetermined motor efficiency and an assumed over-all efficiency for the speed-reducing mechanism of 90 per cent. Brothman (4)illustrated graphically the effect of viscosity on the power requirements of various types of mixing equipment. The reliability of the power data given in the literature depends on the method in which it was obtained and can be assigned four classifications, the first for data of the highest probable accuracy, the second for the next best, and so on: 1. Power measurements made with a dynamometer (6, 7, 1.6, 16). White and Sumerford stated that their data could be used t o estimate power for paddle agitators within *40 per cent. The
accuracy of the other data in this classificationis probably better. 2. Power measurements made by determining the net input t o the drivin motor, acconnting for the efficiency of the motor at the givenqoad (8, 11). Since a speed-reducing gear train and a Reeves adjustable pulley drive were used in the measurements reported by the New England Tank and Tower Company and an over-all efficiency for this mechanism of 90 per cent was assumed the power reported at low speeds should be considered of doubtful accuracy; but the power reported at high speeds, where the driving equipment was operating at or near rated capacity should be fairly accurate. 3. Power measurements based on the difference between no-load and load readings at the same speed (2,9). This method does not take into account the change in efficiency of the driving motor and the speed-reducing mechanism due to change in load. Poorest accuracy should be expected for the results obtained when o erating at a small percentage of the rated motor capacity. 4. 3ower requirements based on calculated pumping capacity (10, 12).
The direct use of an electric motor t o measure power input to stirrers involves many difficulties: (a) The difference between no-load and load readings for small size equipment is sometimes less than the accuracy of the meter readings;
195
FIGURE1. DYNAMOMETER FOR R~EASURING POWER INPUT TO AGITATORS
(b) the fluctuations in line voltage are wmetimes sufficiently large to introduce an enormous error in the net load readings; and ( c ) it is difficult to estimate accurately the power lost due to bearing friction or friction in a speed-reducing mechanism since they may vary greatly depending on temperature, lubrication, load, speed, and time of operation. The dynamometer used in this investigation was designed to overcome these difficulties.
Apparatus and Materials The torquf: table, described in detail by Hixson and Luedeke (7), was modified to act as a dynamometer t o measure the torque required to rotate a stirrer. As illustrated in Figure 1, a framework of angle iron was bolted t o the horizontal circular table. Both the width between the four uprights and the clearance between the upper horizontal crosspieces and the table were made sufficiently large t o accommodate the 45.7-cm. vessel. A slowspeed, direct-current, shunt motor, with vertically adjustable lugs, was bolted to two vertical cylindrical shafts which were, in turn, attached to the two horizontal crosspieces. The motor and framework were symmebrical so that they would produce little horizontp,l thrust on the torque table bearing. The axis of the motor was set t o coincide as closely as possible with the turntable axis. The rotating frame and the torque table were placed on a platform scale, and the whole was set inside a fixed framework with a vertically adjustable platform. The agitation vessel rested on the platform of this fixed framework, and the stirrer was coupled t o the motor on the rotating frame. The wires leading t o the motor were suspended from above so as t o interfere as little as possible with the rotation of the frame. Speed control was obtained by means of an adjustable rheostat in the line t o the motor. The rotational speed of the stirrer was measured with a variable-speed stroboscope by observing a chalk line on the motor coupling. Very low speeds were measured by actual count. The 15.2-, 20.6-, 26.0-, 35.9-, and 45.7-cm. vessels and turbine blades described in the first paper of this series (6) were used in this investigation. The three larger vessels were varnished, and the 15.2- and 20.6-em. vessels were free of any coating. The texture of the surface of the smaller vessels was that of ordinary galvanized iron. No effect of surface roughness was evident in the runs made. In addition to the standard 45" turbine impeller, the 26.0-em. vessel was fitted with a 60' turbine and a 90" (paddle) stirrer, both of the same length and width as the 45" turbine impeller. A series of runs was made with
INDUSTRIAL AND ENGINEERING CHEMISTRY
196
Vol. 34, No. 2
d
.~ h
m
(0
N
?
3
0
m
? m
?
Y
N
N 0
N
2
5
3
X
d : ? 3 %
N . 2 l o o 3 N
9 w W
February, 1942
INDUSTRIAL AND ENGINEERING CHEMISTRY
baffled vessels, and the dimensions of the baffles used are shown in Fi ure 7. Where the 45' impeller was used in a cylindrical vessef with standard clearance, clockwise rotation, no baflles, and a liquid depth equal to the vessel diameter, the apparatus is referred to as the standard design. Runs made under other conditions are indicated as variations from the standard design. The liquids em loyed in this investigation were tap water, 40 per cent sucroseso?ution, 60 per cent sucrose solution, g.lycerol, and two oils of known viscosity and density. The lighter oil is referred t o in Table I as medium, and the heavier oil as heavy, although both were extremely viscous. The sucrose solutions were made from commercial cane sugar by weight, and their specific gravities were checked frequently. A small amount of sodium chloride and fluoride was added to these solutions to preserve them. These salts did not affect the fluid properties to any measurable extent. The glycerol had a purity of 99 per cent or better, and was kept tightly covered at all times when not in use to prevent absorption of moisture. The fluid properties for water, sucrose solutions, and glycerol were all obtained from the International Critical Tables.
197
where c is a dimensionless constant depending on the shape of the body, the shape of the limiting surface of the fluid, and the state of streaming. It is a function of both the Reynolds number (Re) and the Froude number, defined for an agitation system as n d 2 p / p and n*d/g, respectively. However, data on power requirements of agitators are accurately correlated if it is assumed that coefficient c is a function of Re only. The validity of this assumption is confirmed by the data of this investigation and others (6, 7), correlated on the basis of Re only. Applying Newton's law t,o the differentid section
Experimental Procedure With the above design the torque transmitted through the rotating framework to the turntable is essentially that due to the load on the stirrer alone. (Actually an additional torque is transmitted due to windage losses and the inertia of the rotating shaft and coupling, but it was found to be negligible in comparison to the torque due to the agitator load.) The method of balancing this torque and compensating for the friction of the torque table bearing differs slightly from that employed by Hixson and Luedeke: Two pins were attached t o the circumference of the rotating table, approximately 1 inch apart. In any run a sufficient wei ht was added to the weighing pan to balance roughly the torque %e t o the agitator load. The table was then given a slight rotary motion, by tapping, in a direction opposed t o the action due to the weights on the pan, so that at the point where the motion of the table reversed, the fixed pointer was approximately 1/18 inch from one of the moving pins. If the table then rotated back to the second pin, the weight was considered balanced. If not, more or less weight was placed on the pan until this balance was achieved. A similar procedure was followed a t the end of any series of runs, with the stirrer detached, to measure the torque required to overcome the friction of the torque,table bearing. Then the resulting torque for any run was determined by subtracting this last value from the reading under load conditions. Before any particular series of runs, the vertical positions of the motor and adjustable platform were fixed t o secure the proper stirrer clearance. The vessel was filled with liquid to the desired height and placed in the proper position relative to the stirrer. The temperature of the liquid was taken with a thermometer, and this was averaged with the temperature at the end of the series to evaluate the fluid pro erties. The temperature variation from the beginnin to the en{ of any series was no more than a few tenths of a 8egree centigrade. The motor was started and allowed t o come to top speed. Sufficient time wm allowed to elapse until steady-state conditions were attained in the vessel. In the case of water and the less viscous liquids, a few minutes sufficed, but for the more viscous fluids, at least 5 minutes were allowed. The torque table was then balanced in the manner described, and the speed of the stirrer measured immediately afterward. In some runs the vertical thrust on the platform scale was also measured. The rheostat was then adjusted for another speed, and the same procedure repeated. The liquid used, the vessel size, the speed, the torque, the temperature, and the fluid properties are recorded for all runs in the tables.
FIQURE2. ILLUSTRATION FOR DERIVATION OF EQUATION 12
(5)
where V is the relative velocity between the blade and liquid a t a section a distance I from the centerline of the shaft. Assuming that this velocity, V ,is directly proportional to the velocity of the differential section, or V
=
b ( 2 4
(6)
and substituting Equation 6 in Equation 5,
Integrating Equation 7,
T = k'cWnzpL4
(9)
where But for geometrically similar systems, W and L are directly proportional t o some size factor (in this investigation chosen as the vessel diameter), d, and Equation 9 can be written: T = k'cdspnz
(11)
Since k" is a factor of proportionality only, it has been included in the definition of c in this investigation, and the resulting Equation 12 is used:
Power Requirements of Rotating Bodies in Geometrically Similar Systems The method developed for the correlation of the experimental results of this investigation is a modification of those of Hixson and Luedeke and of Buche. It is based on an integration of Newton's law expressing the friction drag resulting from relative motion between a body and a fluid in contact. Newton's law is F = c A -V2P 2g
(3)
Correlation and Discussion of Data I n addition to the series of runs made with the standard design, six variations from this design were studied. The results for each design are discussed under their respective headings below. I n all runs a consistent set of units was used to evaluate c and Re (gram-second-centimeter system). The Reynolds number is a dimensionless group and any other
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
198
Vol. 34, No. 2
IO’
I0 5
2
VESSEL SYMBOL
IO+ 7 5 3 2
+I,i C
V
IO+ 7 5
3
2 3 4 5 6 810* 2 34.56810” 2 3456810“ 2 3 4 5 6 8 Re=&
P
FIGURE3. c
us.
Re
FOR THE
STAXDARD DESIGX
3
&
2
C
IO6 ; 1 7 5 4 3 2 34 6
8ld
2 3 4 68104 2 3 4 6810’ Re=&
FIGURE 4. c vs. Re
FOR
2 3 4 68106 2
I”
REVERSED ROTATION
5 3
4568103 2 3456810“ 2 3456 810’ 2 Re=n d P
2 3 4 5 6 8106 2
P
FIGURE5 . c us. Re
FOR
VARIABLELIQUID DEPTH
c =
DATAFOR GEOMETRICALLY SIMILAR TURBINE AGITATORS TABLE11. POWER WITH REVERBED (COUNTERCLOCKWISE) ROTATION Fluid Water
40v0 sucrose
60% sucrose
d
n
T
t
r
P
0
Re
35.9
4.25 5.25 6.25 7.75 8.16 9.34
439 651 841 1310 1440 1780
24.0
0,00902
1.00
4.09 X 10-7 3.96 3.61 3.66 3.62 3.42
6.07 X 106 7.50 8.94 1.11 108 1.17 1.34
45.7
4.50 5.59 6.54 7.64 8.29
1370 25.0 2040 2780 3760 4330
0.00895
1.00
3.41 3.29 3.27 3.32 3.17
1.05 1.31 1.53 1.76 1.94
20.6
4.62 5.34 6.16 7.11 8.59 9.40 10.3
45.7 24.3 59.4 77.6 98. 1 139 171 194
0.0540
1.18
4.87 4.71 4.68 4.42 4.30 4.42 4.20
4.29 X 10‘ 4.95 5.71 6.61 7.96 8.73 9.54
26.0
4.41 4.87 5.81 6.84 7.25 10.1 8.33 9.11 10.1
146 162 215 274 308 571 386 502 546
25.0
0,0620
1.18
5.41 4.86 4.38 4.17 4.17 3.94 3.96 4.30 3.82
6.76 7.49 9.07 1.05 X 10‘ 1.11 1.56 1.27 1.40 1.55
15.2
6.46 7.87 9.16 10.5
34.3 45.7 70.8 86.8
24.0
0.460
1.29
7.82 7.05 8.07 7.50
4.19 X 108 5.09 5.94 6.80
20.6
4.27 5.16 6.12 7.00 7.46 8.59 9.50 10.3
73.1 89.1 126 160 176 231 281 327
24.3
0,452
1.29
8.40 6.99 7.05 6.80 6.60 6.54 6.54 6.47
5.17 6.25 7.41 8.47 9.04 1.04 1.15 1.25
4.25 5.21 6.21
183 258 382
24.2
6.60 6.18 6.47
8.16 X 101 1.00 x 104 1.19
4.75 5.25 6.34 7.26 8.29 9.50 10.3
70.9 73.1 77.7 93.6 114 148 171
23.8
4.34 4.75 5.41 6.87 7.91 9.25 10.2
160 180 219 322 400 507 567
23.8
4.34 5.91 6.84 7.41 8.91 9.87
404 662 820 939 1250 1470
24.0
26.0
Glycerol
199
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
February, 1942
15.2
20.6
26.0
0.455
1.29
1.26
5.8
5.8
1.26
5.7
1.26
3.09X 2.60 1.90 1.74 1.63 1.61 1.57
lo-*
239 264 318 364 416 476 619
1.82 1.71 1.60 1.46 1.37 1.27. 1.17
400 437 499 634 730 851 941
1.43 1.27 1.17 1.14 1.05 1.01
649 883 1020 1110 1330 1480
consistent set of units can be used to evaluate it. The group (T/d5pn2) is not dimensionless, and caution should be observed in transposing i t from one set of units to another. STANDARD DESIGN. The results (Table I) for six different liquids in five different sized vessels over a speed range of 120-600 r. p. m. are correlated in Figure 3. The striking similarity between this curve, the well-known friction factor curve, and the curve correlating data for free-falling spheres (IS)is a t once evident. At low Reynolds numbers the data extrapolate to a straight line with a slope of 1.0. This fact could be predicted from Equation 12 to satisfy the conditions for rotating viscometers (such as the modified Stormer viscometer, 12) where
-
x
x
B
(A)
in order that the torque be directly proportional to the speed and viscosity. This curve also illustrates the importance of limiting the size factor, d, and speed, n, of rotating viscometers so that the resulting Reynolds number is within the laminar region. A comparatively long transition zone follows the laminar flow region where the slope of the curve gradually changes. Within this transition zone a break in the curve occurs, corresponding to the transition from streamline to turbulent flow in conduits. This region where the break occurs has been analogously termed the “critical region”. The critical value found for the Reynolds number in this investigation (approximately 5 X 108) agrees well with the value found by Hixson and Luedeke (approximately 6 X lo3) who measured wall friction in similar equipment. Beyond the critical region, the curve flattens out to a constant slope of -0.12. Above Re = 4 X 104, the equation of the straight line shown is
104
or T = 1.82 X
10-4n1.88d4.76p0.88~0.12(14)
and the power, in terms of horsepower, is found by multiplying T (in gram-em.) bY 6.2%
454 X 2.54 X 12 X 550
=
8.25 X 10% (15)
and the resulting expression is p
1.50 X
10-10n2.88d4.78p0.88~0.12
(16)
in which the exponents agree well with those of Equation 1. For predicting the power requirements a t Reynolds numbers below 4 X 104, the actual curve should be used to estimate the value for c. This curve also illustrates the importance of selecting designs of sufEciently large size and liquids of suitable properties when model experiments are to be used for predicting the power requirements of large size equipment. Unless the range of Re for the models lies on the flat portion of the turbulent flow curve, extrapolated values will be erroneous. REVERSEDROTATION. I n this series of runs (Table 11) the direction of the impeller was reversed so that the blades forced the liquid down toward the bottom of the can rather than up as in the standard design. All other conditions were the same. A ourve, similar to that in Figure 3, was obtained as shown in Figure 4. I n this and the succeeding figures, a dotted line is drawn for the standard design t o compare the effect of the design variation on the power requirements.
200
INDUSTRIAL AND ENGINEERING CHEMISTRY
By comparing the solid and dotted lines in Figure 4, it is evident that the reversed rotation causes a lowering in the power consumption a t low Reynolds numbers, but a t high values the curves converge and the power requirements are the same, independent of the direction of rotation. LIQUID DEPTH VARIABLE.I n one set of runs in this series (Table 111) the liquid depth was made half the vessel diameter (h/d = 0.5), and in the second set the liquid depth was made three quarters of the vessel diameter (h/d = 0.75). The points for both sets are shown in Figure 5 where the upper curves should be used with the right-hand scale and the lower curves with the left-hand scale. When any of the cans used was only half filled with a low-viscosity liquid, and the impeller was rotated a t a comparatively high speed, the resulting vortex was sufficiently deep to uncover the blades. As a result, the torque would not increase with increasing speed a t the same rate as would ordinarily be expected, and a plot of the experimental points would show that, for a vessel of any particular size, the data for high speeds branch away from the curve correlating the results for low speeds. This phenomenon is most noticeable for low-viscosity fluids in the large vessels where high peripheral speeds are attained. As the viscosity increases and the size decreases, the effect of vortex formation almost completely disappears. The same effect is indicated by the experimental results for the vessels three fourths full, but since the vortex formed here cannot reach the blades so easily because of the liquid depth, the effect is less pronounced. In Figure 3 this phenomenon has entirely disappeared when the liquid depth is equal to the vessel diameter. As would be expected, lowering the quantity of liquid in the vessel lowers the power requirements at a given Reynolds number, but this effect disappears a t low Reynolds numbers where the curves converge with the curve for the standard design. The difference between the power requirements for a vessel half full compared to a vessel three quarters full is not great. IMPELLER PITCHVARIABLE. Two sets of runs were made in this series (Table IV), both with the 26.0-cm. vessel. I n one set the impeller with 60" blades was used, and in the second set a four-blade paddle was used. The results are plotted in Figure 6 with the line correlating the results for the 45' impeller in the standard design. All curves seem to converge a t low and high Reynolds numbers, but a t inter-
Vol. 34, No. 2
TABLE111. POWER DATAFOR GEOMETRICALLY SIMILAR TURBINE AQITATORSWITH LIQUIDDEPTHVARIABLE Fluid Water
d
n
T
t
P
P
35.9
45.7
2.00 2.45 2.70 2.53 2.17 1.75 4.30 4.16 5.12 6.25 7.50 8.91 9.87
91.4 110 146 128 100 68.6 345 354 436 585 656 73 1 804
28.0
0.00837
23.2
0.00931
1.88 2.07 2.25 2.42 2.72 4.29 6.25 5.37 6.91 8.09 9.35
274 28.0 319 358 422 491 1110 24.0 1520 1380 1560 1900 2040
0.00837
0.00902
Re
0
Liquid Depth = '/n Tank Diameter ( h / d 26.0 10.3 171 26.8 0.008S1 1.00 8.97 146 8.00 162 9.08 158 10.4 171 7.53 144 6.03 128 5.00 91.5 4.37 68.6 10.5 187 8.58 178 2.38 27.4 2 5 . 0 0.00894 1 . 0 0 2.13 22.8
= 0.5)
1.36 X 10-7 1.52 2.13 1.61 1.33 2.13 2.96 3.08 3.02 1.42 2.03 4.06 4.21
7.91 6.89 6.14 6.97 7.99 5.78 4.63 3.84 3.36 8.25 6.75 1.80 1.61
x
105
x
io8
1 . 0 0 3.83 3.07 3.35 3.35 3.56 3.81 3.13 1.00 3.43 2.79 2.51 1.96 1.54 1.58
3.08 3.77 4.16 3.90 3.34 2.70 6.62 5.75 7.06 8.62 1.03 1.23 1.36
1.00
3.90 3.74 3.56 3.62 3.34 1 . 0 0 3.04 1.96 2.40 1.64 1.46 1.18
4.70 X 106 5.17 5.62 6.05 6.80 9.95 1.45 X 103 1.25 1.60 1.88 2.16
40% sucrose
26.0
4.59 5.25 6.46 7.34 9.34 10.2
110 148 208 247 279 300
24.2
0.0540
1.18 3.73 3.84 3.55 3.26 2.28 2.06
6.76 X 10' 7.75 9.55 1.08 X 106 1.38 1.50
60% sucrose
15.2
5.47 8.33 10.0
20.6 57.1 70.9
25.6
0.423
1.29
6.56 7.87 6.77
3.86 X 108 5.87 7.04
1.29
7.37 7.40 5.94 5.71 5.77 5.50 5.20
5.65 6.63 8.00 9.25 1 . 1 3 X 104 1.24 1.30
1.29
7.19 1.02 5.63 5.44 5.25 4.98 3.73
26.0
Glycerol
Water
66.2 41.1
28.2
0.430
4 . 9 8 X 108 3.29 8.73 1 . 1 3 X 10' 1.34 1.63 2.03
2.45 1.62 4.30 5.58 6.58 8.03 9.97
260 349 494 569
20.6
4.20 4.83 5.75 6.70 7.95 9.25 10.2
151 199 242 292 356 457 523
26.2
4.50
1.26
1.83 X 1 0 - 8 1.82 1.57 1.39 1.20 1.14 1.07
500 575 684 797 946 1100 1210
26.0
2.00 4.17 5.08 6.63 8.20 9.80
116 368 477 729 994 1340
25.8
4.70
1.26
1.93 1.44 1.23 1.10 9.96 X 10-7 9.30
371 774 941 1230 1520 1820
0.75) 3.40 X 10-7 3.28 3.08 3.20 3.12 2.96 2.64 2.49
1.76 X 105 3.39 3.91 4.70 5.41 6.26 7.30 7.98
3.53 3.48 3.29 3.06 3.05 2.30
6.17 6.83 8.39 9.85 1.11 x 10s 1.35
160
-
Liquid Depth a/4 Tank Diameter ( h / d 26.0 2.25 20.5 26.5 0.00865 1.00 4.33 73.2 91.5 5.00 137 6.00 178 6.92 226 8.00 274 9.33 10.2 308 35.9
4.41 4.87 6.00 7.04 7.91 9.66
409 23.7 494 704 902 1140 1310
0.00821
1.00
-
x
x
10-0 10-7
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
February, 1942
TABLE 111. POWER DATAFOR GEOMETRICALLY SIMILAR TURBINE AGITATORS WITH LIQUID DEPTHVARIABLE(Continued) Fluid
d
T
n
Liquid Depth
t
p
p
Tank Diameter ( h / d
-
Re
c
Water (oont'd)
45.7
4.25 5.04 6.25 7.35 8.75
0.75) (Cont'd) 1180 23.0 0.00937 1.00 3.29 1690 2230 2.88 3.14 2820 2.64 3190 2.10
40% aucro8e
26.0
4.34 5.25 6.66 5.66 7.16 8.12 9.12 10.0
110 160 251 183 283 368 448 491
16.2
7.42 8.83 10.2 5.00 5.75 7.20 6.20 7.67 8.17 9.42 10.3
48.0 25.6 0.426 61.6 77.6 16.0 25.0 0.440 25.1 45.7 32.0 50.2 50.2 66.2 73.1
1.29 8.31 7.56 7.14 1.29 6.11 7.25 8.41 7.94 8.15 7.18 7.11 6.59
5.22 X 101 6.21 7.17 3.39 3.89 4.87 4.20 5.20 5.53 6.38 6.97
20.6
6.50 7.83 8.37 9.26 10.1
160 206 247 296 330
1.29
26.0
1.85 2.50 4.33 5.68 7.13 8.55 9.92
46.7 26.0 0.440 91.4 206 316 439 633 825
7.91 7.90 7.35 7.19 6.74 1.29 8.70 9.52 7.14 6.60 5.62 5.64 5.45
8.35 8.42 1.08 x 104 1.19 1.30 3.66 X 101 4.95 8.57 1.10 x 104 1.41 1.69 1.97
20.6
4.28 6.20 6.33 7.17 9.08 9.95 8.70
160 212 268 322 450 644 434
26.6
4.40
1.26 1.87 X 10-8 521 1.68 632 1.43 770 1.34 871 1.17 1100 1.17 1210 1.23 1060
26.0
2.52 1.76 4.83 6.42 7.97 9.63
I99 25.8 123 732 474 1030 1370
4.70
1.26 2.09 2.68 1.18 1.35 1.08 9.85 X 10-7
60% sucrose
Glycerol
a/4
25.0 0.0520
25.5 0.426
9.48 X 101 1.40 1.12 x 108 1.64 1.95
1.18 4.16 4.29 4.02 4.07 3.93 3.97 3.50 3.83
6.66 X 8.06 1.02 x 8.69 X 1.10 X 1.25 1.40 1.54
467 825 896 1180 1480 1790
104 106 104 106
20 1
on the fluid in an agitation vessel is to impart a horizontal motion, and Equation 2 indicates the power used in overcoming the horizontal component of wall friction. BAFFLED VESSHLS. Baffles were used in the 15.2-, 26.0-, and 45.7-cm. vessels in this series of runs (Table V). The addition of even one baffle to a circular agitation vessel materially affects the nature of the flow by destroying the vortex present. The correlation shown in Figure 7 is for vessels containing two and four baffles, and apparently the results obtained are independent of the number of baffles present. I n the laminar flow region, the addition of baffles has little effect on the power requirements. No runs were made for the baffled vessels in a range of Reynolds numbers corresponding to the critical region for the standard design, but the data for the transition zone and turbulent region are both fitted with a smooth curve which would seem to indicate the absence of a critical region for baffled vessels. Above Re = lo4 the baffled vessels consume a large amount of power compared to the unbaffled vessels, and the curves diverge sharply. For some inexplicable reason, the data for water in the 45.7-cm. vessel did not coincide with data for the smaller vessels. This is illustrated by the two separate curves shown at Re = 5 X lo6, the upper line correlating the results for the 45.7-om. vessel. This phenomenon was checked several times, and the same result was inevitably obtained. The lower line seems to in-
mediate values the 60" and 90' impellers require more power. Above Re = 105 the slope of the line for the paddle agitator is -0.23, and the equation of the straight line for turbulent flow is m
and in terms of horsepower,
P = 10.1 x 10-10n%77d4.64p0.?7M0. 28 (18) The exponents are in excellent agreement with those of Equation 2. This might be expected since the action of a paddle
2 34568102 2 3456810' 2 3456810" 2 3456810' Re. nd$ P IMPELLER PITCH FIQURE6. c os. Re FOR VARIABLE
INDUSTRIAL AND ENGINEERING CHEMISTRY
202
DATAFOR GEOMETRICALLY SIMILAR TURBINE AGITATORS TABLE IV. POWER WITH IMPELLER PITCHVARIABLE Fluid
d
n
T t Blade Angle
-
P
RL
c
P
60°
Water
5.03 4.83
6.69 7.39 8.12
1.18
8.65 8.33 8.33 7.69 7.69 7.69 7.44 7.24 7.05
6-08 104 7.33 8.30 9.93 1.07 X 106 1.08 1.23 1.29 1.36
2.25 2.17 2.05 1.97 1.89 1.79 1.79 1.74 1.72 4.32 4.20 3.97 3.48 3.52
632 703 800 931 1020 1180 1240 1320 1410 150 164 182 222 242
0% sucroLie
26.0
4.25 5.16 5.84 7.00 7.54 7.59 8.62 9.09 9.59
219 311 400 62 9 613 620 775 841 910
23.0
0.0561
60% sucrose
26.0
4.50 5.00 5.71 6.66 7.75 8.12 9.00 9.50 4.38 4.00 5.04 5.62 6.60 7.50 8.60 9.00 9.50
425 498 624 782 990 1120 1360 1500 471 391 581 695 940 1190 1520 1660 1840
23.8
0.470
9.3 9.2 9.1 8.8 8.6 8.3 6.4 8.0 7.7
1.23 1.24 1.26 1.30 1.33 1.38 1.36 1.42 1.48
4.16 4.62 5.26 6.12 6.71 7.79 8.16 8.71 9.25 4.41 4.91 5.59 6.84 7.92
601 696 850 1110 1280 1630 1790 1980 2200 1270 1530 1870 2460 3320
24.0
5.60
1.26
7.9 7.7 7.5 7.4 6.8
25.3 25.8 26.3 26.5 28.0
1.27
Glycerol
26.0
x
Blade Angle = 90' (Paddle Type Impeller) 1.00
Wster
6.37 X 10-7 5.84
3.78:X 4.56
o.au 5.70 5.65 4.99
6.66 7.42 8.20
1.03 X 10-8 Q.24 X 10-7 8.90 8.78 8.46 8.20 7.82 7.11
6.26 X IO' 7.62 9.15 9.81 1.14 X 1 O b 1.21 1.35 1.44
4.14 3.58 3.34 3.50 3.31
2131
P.62
40% sucro8e
26.0
4.41 5.37 6.43 6.91 8.04 8.50 9.50 10.1
281 23.0 375 516 590 764 832 990 1020
0.0661
60% sucrose
26.0
4.37 4.79 5.35 6.25 6.76 7.85 8.80 9.34 4.46 6.12 5.75 6.59 7.04 8.06 9.16
452 537 651 845 994 1290 1610 1770 655 801 895 1200 1380 1710 2160
23.5
0.471
10.9 10.8 10.6 10.3 10.0 9.8 9.6
1.04 1.05 1.07 1.10 1.14 1.16 1.19
4.16 4.66 5.41
742 903
24.6
5.30
Glyoerol
26.0
8.18
7.12 7.66 8.34 9.00 4.50 5.00 0.34 7.00 7.79 8.59
1.16
5.39 a.
It)
1250
1520 2040 2370 2660 3060 1280 9 . 6 1560 9.3 2170 9 . 0 2470 8 . 9 3200 8.7 3890 8 . 3
--
21.5 22.1 22.7 23.0 23.3 24.2
106
194 240
288 305
Vol. 34, No. 2
crease izraduallv in sloDe. and it may be thai a t higher Reynhds numbers the two curves will coincide. For prediction by extrapolation of the power requirements of vessels larger than 45.7 cm. in diameter, it is recommended that the upper curve be used since it will indicate safe-side values for horsepower. As will be shown later, the upper curve can be used to predict power data accurately for vessels 12 feet in diameter. VARIABLE CLEARANCE. I n this group two sets of runs were made (Table VI) : one with the impeller fairly close to the tank bottom ( y / d = 1/12), the other with the impeller a t a greater clearance from the tank bottom ( y / d = I/s) than in the standard design. The results are plotted in Figure 8. With low clearance the power requirements are increased appreciably. Surprisingly, when the clearance is greater than standard, the power requirements again increase. Apparently there is some optimum clearance for lowest power consumption, although a t high Reynolds numbers this effect disappears and the curves converge. Some data of the New England Tank and Tower Company on the power requirements of a propeller agitator, for various values of clearance, confirm the observation noted as t o an optimum clearance for minimum power. For high values of clearance in thin liquids, the vortex is sufficiently deep aisome speeds to uncover the impeller blades, and the same branching of data noted in Figure 5 takes place. VARIABLE IMPELLER SIZE. To studv the effect of maintaining the vessel sizi constant and varying the impeller size, the 35.9-cm. vessel was used in conjunction with the impellers from the 20.6-cm. vessel, the 26.0-cm. vessel, and the 45.7-cm. vessel (Table VII). The data for all three sets are indicated in Figure 9 in terms of the ratio of the blade area of the impeller used to the blade area of the standard impeller for the 35.9-cm. vessel (AIA,),and in terms of the ratio of the vessel diameter t o the diameter of the blade (djd,). Each set of data is correlated by a line which is approximately parallel to the line for the standard design ( A / A , = 1.00 and d/d, = 3.00). All data are limited t o the turbulent flow region. Figure 10 was drawn to aid in estimating the value of c for impellers of sizes other than those studied. The main factors in determining the power consumption of an impeller are the area of the blades and the location of the area, in respect to the center of rotation. Hence the recommended procedure for evaluating c for blades different in size from those studied is
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1942
2
.Ik
C
V
IO 7 5
3 2 IO
FIGURE7. c us. Re
FOR
BAFFLED VESSELS
4 3 2
3
456 8103 2 3456 810" 2 3 4 6 810' 2 3 4 68106 2 Re. FIGURE8. c us. Re
FOR
d e P
VARIABLE CLEARANCE
11
U
68104 2 3456810' 2 34568106 2 34568 Re =- n d2P
P
FIGURE9. c us. Re
FOR
VARIABLE IMPELLER SIZE
203
INDUSTRIAL A N D E N G I N E E R I N G C H E M I S T R Y
204
based on recognition of these two factors: Evaluate terms A / A , and d / d a from the measured surface dimensions of the turbine being considered. From the value of d / d b and Re, find the value of c from the upper graph of Figure 10; from the value of A / A , and Re, find the value of c from the lower graph. The average of these two values of c should be used to approximate the power requirements. The validity of this method is confirmed by the examples shown in a subsequent section of this paper.
TABLE v. Fluid
d
Water
T
t
p
P
Re
C
40% aucrose
5.34 6.35 7.25 5.59 9.46 10.3
13.7 24.5 18.3 22.5 32.0 38.8 52.5
0.00903
1.00 5.94 X 10-3 1.36 X 105 5.64 1.62 5.35 1.86 6.35 2.20 5.34 2.42 6.06 2.65
26.0
4.50 5.34 5.66 7.25 8.25 9.09 9.91
151 217 233 425 519 649 786
0.0105
1.00
6.26 6.41 6.11 6.80 6.41 6.60 6.73
2.90 3.45 3.65 4.67 5.32 6.86 6.39
45.7
4.58 4.75 5.16 5.84 6.59
3500 7.0 3880 4590 5690 7120
0.0142
1.00 8.40 5.65 5.65 8.40 8.26
6.75 7.00 7.60 8.60 9.69
26.0
4.25 5.00 6.25 6.97 7.59 9.00 9.91
155 224 336 420 505 706 575
26.3 0.0503
1.18 6.14 6.41 6.14 6.17 6.25 6.23 6.35
6.75 X 10' 7.95 9.94 1.11 x 101 1.21 1.43 1.58
15.2
4.56 5.00 6.66 6.79 8.17 9.04 10.3
48.0 26.5 61.6 70.8 93.6 119 142 171
20.0
60% sucrose
Glycerol
4.50
1.26
2.26 X 10-8 295 2.41 324 2.16 366 2.03 439 1.75 528 1.71 584 1.56 669
Four Vertical Baffles Water
4.34 4.66 5.71 6.66 7.71 9.16 10.0
162 169 266 393 477 649 787
4.34 4.91 5.54 6.21 4.29 4.75 5.25 6.79 6.50 1.68 2.08 2.27 2.48 4.30 4.75 6.20 5.80 6.47
3840 7.5 0.0141 1.00 9.49 6.44 4a20 9.04 7.29 8.96 8.20 6490 7210 9.35 9.20 9.10 3810 21.0 0.00985 1.00 9.04 5.96 1.01 x 101 4020 1.11 8.71 4770 1.23 5760 8.65 1.38 8.77 24.7 0.00896 1.00 1.07 X 10-8 3.92 X 105 9.52 x 10-7 4.55 821 9.46 5.29 970 9.52 5.79 1165 9.27 1.00 x 106 3410 1.11 9.26 4180 1.21 9.I7 4840 1.36 8.91 5960 1.51 8.72 7270
g!:o
15.2
4.41 5.25 6.16 7.50 8.91 10.3
50.3 26.4 64.0 75.4 112 123 158
26.0
4.34 5.00 5.54 6.54 7.69 8.66 9.59
398 26.0 4.60 507 603 763 980 1200 1420
26.0
45.7
Method for Predicting Power Requirements If the contemplated design of an agitator agrees in all respects with one of the designs studied in the present investigation, the c vs. Re plot for that particular design can be used to predict the power requirements for the largescale equipment. The contemplated design may, however, differ in more than the one way covered by the curves in Figures 4 to 10. For example, baffles may be desired in a half-filled tank with an enlarged impeller. This design would vary in three respects, instead
n
15.2
Surging When the larger vessels were used with low-viscosity liquids and without baffles, surging took place at some speeds. This would start out as a slight wave motion on the surface of the liquid which would gradually increase in intensity to a constant value, but in some cases it became violent enough to splash liquid from the tank. There was no way to predict when surging would occur, but it seemed to take place most frequently at the lower (150-250 r. p. m.) rather than a t higher speeds. The torque reading on the turntable was not influenced by moderate surging. Since the effect of surging is to reduce the capacity of the tank, it is important to eliminate it. We found that the addition of even one vertical baffle to a cylindrical vessel eliminates any tendency to surge and also destroys any vortex present.
DATAFOR GEOMETRICALLY SIXILAR TURBINE AGITATORS I N BAFFLED VESSELS TNOVertical Baffles
Vertical Thrust The vertical thrust created by the action of the turbine was measured in some runs by reading the total weight on the platform scale and subtracting the weight when the agitator was at rest. The vertical thrust was comparatively low and could not be determined with much accuracy since it involved a small difference between two large weights. For the 45.7-cm. standard design, using water, the vertical thrust was only 2 pounds a t 500 r. p. m.
POTER
Vol. 34, No. 2
Glycerol
24.0 0.00915 1.00 7.25 6.53 6.60
6.46 6.73 6.46 6.73
4.50
1.26
3.20 3.44 4.22 4.92 5.70 6.76 7.40
2.53 X 10-8 285 2.28 340 1.96 399 1.95 485 1.52 576 1.47 662
1.26 1.41 1.36 1.26 1.19 1.10 1.06 1.03
800 925 1060 1210 1430 1600 1780
INDUSTRIAL A N D ENGINEERING CHEMISTRY
February, 1942
TABLE VI. POWER DATA FOR GEOMETRICALLY SIMILAR TURBINEAGITATORS WITH VARIABLE CLEARANCE Fluid
d
n Clearance
Water
60% sucroee
T
-
t 1/19
C
25.0
Re
1/11)
0.00895 1.00 5.42 6.42 X 10-7 3.97 3.15 X 105 5.41 4.78 5.21 5.39 4.95 6.14 4.72 6.80 4.52 7.45
4.17 5.25 6.33 7.13 8.13 9.00 9.87
133 178 258 315 390 455 525
35.9
2.02 4.33 4.75 5.25 6.17 7.25 8.13 9.20
119 25.7 466 549 674 878 1170 1700 1480
0.00881 1.00
45.7
1.60 2.22 2.77 4.58 4.17 5.20 6.17 7.13 8.08
345 525 8.8 762 1480 1810 2150 2970 4000 4610
20.6
4.33 5.30 6.42 7.17 8.17 9.03 9.92
144 178 226 265 331 404 463
26.0
2.27 4.17
121 338
4.90 4.17 4.07 4.10 3.87 3.73 3.37 3.75
2.95 6.32 6.94 7.66 9.01 1.06 X 101 1.34 1.19
0.0136
1.00 5.35 6.77 5.00 4.27 4.33 4.00 3.91 3.94 3.55
2.46 X 106 3.42 4.26 6.41 7.05 8.01 9.50 1.10 x 106 1.24
25.2
0.431
1.22
25.5
0.427
6.30 5.25 7.17 8.20 9.13 9.58
476 612 848 1050 1250 1380
1.22 1.62 1.34 1.19 1.10 1.13 1.07 1.03 1.03
4.39 x 101 8.05 1.20 x 10' 1.01 1.38 1.58 1.76 1.85
2.08 4.25 4.67 5.17 6.17 7.25 8.25 9.20
190 26.0 551 599 736 943 1220 1480 1880
4.60
1.26
385 786 865 957 1140 1340 1530 1700
26.0
Water
26.0
4.17 5.00 5.97 6.87 7.37 8.80 9.25 10.1
35.9
2.02 4.33 5.33 6.30 7.33 8.45 9.50
20.6
4.53 8.47 6.53 7.33 7.92 9.25 10.1
114 153 171 205 238 313 361
25.2
26.0
2.08 4.28 5.33 6.13 6.92 8.00 8.92 9.92
26.0
2.13 4.28 4.87 5.92 0.92 7.92 8.37 9.47
Clearance
Glyoerol
-
P
26.0
Glycerol
60% sucrow
P
Vessel Diameter ( ~ / d
-
'/a
Vessel Diameter (y/d
84.5 25.0 105 174 233 274 333 358 384
1.70 X 10-6 5.20 X 101 1.40 6.36 1.21 7.70 1.14 8.60 1.09 9.80 1.10 1.08 x 104 1.04 1.19
-
2.93 2.03 1.83 1.83 1.65 1.54 1.45 1.48 I/:)
0.00894
2.93 6.28 7.72 9.14 1.06 X 106 1.22 1.38
1.00
4.81 3.68 3.81 3.51 3.47 3.08 2.68
0.431
1.22
1.23 X 10-8 5.44 X 101 1.13 6.56 8.86 X 10-7 7.83 8.42 8.80 8.37 9.50 8.07 1.11 x 104 7.80 1.21
68.6 25.7 199 324 400 511 671 802 9s5
0.422
1.22
1.09 x 10-1 4.06 x io* 7.49 x 10-7 8.34 7.85 1.04 X 104 1.20 7.33 1.35 7.34 1.56 7.21 1.74 6.93 1.94 6.89
155 25.9 457 537 730 931 1100 1230 1520
4.60
1.26
2.28 X 10-6 1.67 1.51 1.39 1.29 1.17 1.17 1.13
394 791 901 1100 1280 1470 1550 1750
205
of one, from the standard. An empirical method was developed to account for cases of this type: The value of c at various Reynolds numbers was determined from the curves of Figures 4 to 9 and tabulated with the value of c for the standard design at the same Reynolds number, taken from Figure 3. Then the correction factor by which the standard design coefficient had to be multiplied to give the value of c a t the same Re for any other design was evaluated. These correction factors were then plotted for the various designs as shown in Figure 11, where an exaggerated vertical scale is employed to facilitate estimation of the correction factor. The lines correlating the data at low speeds, shown in Figure 5 for power data at variable liquid depth, were used t o calculate the correction factors for all speeds if baffles are present in the vessel, since the effect of baf€ling is t o destroy the vortex which causes the branching of the data as shown. The method for applying the correction factors is illustrated by the examples shown in Tables VI11 and IX based on the data of the New England Tank and Tower Company (11). The agreement between the calculated and measured horsepowers a t low speeds is poor but improves with increased load and speed. The probable reason for this disagreement at low speeds was discussed under item 2 in the accuracy classification of power data (page 195). The same type of calculation is apolicable when the size of the imDeller Is decreased, all other conditio& remaining the same. This is illustrated by the example shown in Table IX. Qualitative observation of the action produced by the stirrers indicated roughly that a t Reynolds numbers above the critical, both the turbulence produced in the vicinity of the impeller and the top-to-bottom turbulence were great. But as the viscosity of the fluid was increased (or the Reynolds number decreased), the over-all turbulence became small and the power consumed by top-to-bottom turnover was negligible compared to the power going to the local action of the impeller. For glycerol and the very viscous oils, little surface action was visible in the range of Reynolds numbers studied. From this, the logical assumption can be made that for very viscous fluids, the factors influencing the power consumption are those due to the rotating element only (i. e., its size, direction of rotation, and clearance), and the power requirements are independent of the size and design of the agitation vessel. This assumption is verified by the torque data for the
Vol. 34, No. 2
INDUSTRIAL AND ENGINEERING CHEMISTRY
206
power consumption. That the vessel design has little influence on power requirements in the viscous flow region is indicated in Figure 7 , where the curves Re Fluid d n T t P P C for both baffled and unbaffled vessels d/db 5.21, A / A a = 0.330 coincide fairly closely for low values of Water 35.9 4.37 34.3 22.0 0.00960 1.00 3 . 0 1 X 10-8 5 . 8 7 X IO5 Re. The example illustrated in Table X 2.97 7.16 5.33 6.53 73.1 50.3 2.88 8.77 further verifies this hypothesis. 7.30 96.0 3 .02 9.80 The excellent check between the cal7.83 105 2.87 1 . 0 6 x 106 8.67 130 2.90 1.16 culated and measured values for horse10.0 185 3.10 1.34 power confirms the assumption made 4.17 5 2 . 5 24.0 0,0543 1.18 4.20 1 . 1 7 X 106 above. This assumption should not be 40% sucrose 4.70 59.4 3.82 1.32 used unless conditions are such that the 5.83 8 0.0 3.34 1.63 6.75 114 3.56 1.89 calculated Reynolds number has a value 3.39 2.06 7.33 128 less than the critical. 8.75 171 3.18 2.46 9.20 194 3 .. 32 86 2.58 2.80 A comparison between the power re10.0 238 3 quirements calculated from the results 60% 8ucros1 4.17 4 . 6 24.1 0.460 1.29 6.33 1.51 of the present investigation and the 5 . 0 8 8123 6.19 1 . 8 4 x 104 8.17 164 5.59 2.24 power measurements of Hixson and 5 . 3 4 2.42 6.67 183 Wilkens for similar equipment showed 7.80 238 5.09 2.82 8 . 5 8 286 5.01 3.11 3.41 erratic agreement. When the sizes and 9.42 322 4 .72 speeds used by these authors were large enough so that the agitator was operatd/db 4.13. A / A i 0.525 ing a t or near the rated capacity of the 35.9 4.17 9 6 . 0 22.0 0.00960 1.00 9 . 2 5 X 10-8 5 . 6 1 X 10' Water 4.75 116 8.60 6.38 driving motor, the agreement was good. 5.33 132 7.77 7.16 The results obtained a t low speeds or for 7.56 8.56 6.37 183 6.70 226 8 .43 0.00 small-size equipment, when the net in8.16 1.06 X 10' 7.87 302 8.67 363 8.09 1.17 put to the driving motor was small, show 8.16 1.33 9.87 475 poor agreement with the values ob9.03 377 7.i5 1.21 tained in the present investigation. 9.92 455 7.7s 1.33 This is to be expected since with the 4.33 151 24.5 0.0635 1.18 1 . 1 4 X 10-7 1 . 2 3 X 10' 40T0 sucrose 6.33 222 1.11 1.52 method of measurement employed by 6.08 275 1.06 1.73 Hixson and Wilkens, the results fall in 6.92 361 1.07 1.97 8.08 441 9.61 X 2.29 classification 3 as far as probable ac8.25 473 9.87 2.34 curacy is concerned. 8.67 544 1 . 0 3 X 10-7 2 . 4 6 9.58 629 9 . 7 4 x 102.72 The effect of surface roughness of the 4.42 238 24.3 0.455 1 . 2 9 1.58 X 10-7 1.62 X 101 vessels has been omitted in this study. 60% suorose 6.00 286 1.49 1.83 This factor apparently has little or no 1.46 2.20 0.03 409 7.09 546 1 . 4 2 2 . 6 8 influence on either the correlation of re1.2s 2.05 8.08 640 8.50 727 1.31 3.10 sults or the prediction of power for the 1 . 2 9 3.48 9.53 905 large vessels. Since the ratio of wetted area to volume for a series of geometrid/db 2.35, A / A s = 1.62 cally similar cylindrical containers is in2.25 411 23.0 0.00936 36.9 Water versely proportional to the diameter, 4.17 1160 5.00 1580 the effect of surface roughness will be 5.58 1900 6.42 2560 much smaller for large vessels than for 3450 7.50 small. 3840 8.17 I n conclusion, the authors wish to 0.0605 1.67 396 20.7 104 40y0 sucrose point out that the power requirements of 2.07 541 2.20 576 an agitator should never be used to 2.35 691 4.33 1890 10' evaluate its mixing efficiency. Buche (6) 4.80 2300 showed that a relation existed between 3200 6.80 3770 6.33 the dissolution constants, measured by 4340 6.83 7.83 4980 Hixson and Wilkens for the svstem benzoic acid-water, and the term nY2, 60% 6uorose 2.03 742 20.5 0.547 1.29 2.34 6 . 1 7 X 10 2.40 1050 2.37 7.30 representing the power input to a series 4.25 3020 2.17 1.29 x 104 of dimensionally similar agitators. Cor4.83 3720 2.07 1.47 2 1.99 . 0 6 1 1.79 . 6 1 relations of this type are misleading since 5.87 5.30 4430 5300 6.53 6510 1.98 1.98 they do not tell a complete story. As an example, in viscous-liquids the dissolution constant decreases severalfold and the pover requirements increase enormously, while Buche's correlation indicates an increasing K with increased power input Until more is known about half-full and three-fourths-full vessels (Figure 5) where rethe true nature of the fluid motion in an agitated container, moval of liquid from a tank a t high Reynolds numbers resulted it is best to treat power input and agitation efficiency as two in a drop in power requirements, while removal of liquid separate problems, considering both when evaluating the from a vessel in the viscous-flow region caused little drop in optimum design but never using one to predict the other. TABLEVII. POWER DATAFOR GEOMETRICALLY SIMILAR TURBINE AGITATORS WITH VARIABLE IMPELLER SIZES 3
-
-
5
Y
I
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1942
Nomenclature
7.0 6.0 A
207
S.0 4.0
2 3.0 8.0
3.0 2.0
FIGURE 10. PLOTSFOR ESTIMATING THE EFFECT OF BLADE LENGTHAND AREAON VALUESFOR c AT VARIOUSVALUES OF Re
A = effective b l d e area, sq. cm. A , = effective blade area for standard design = d2/18, sq. cm. c = drag coefficient = T/dSpna d = vessel diameter, cm. D = vessel diamter, ft. F = drag, grams = acceleration of gravity, cm./sec./sec. = depth of liquid in vessel, cm. H = depth of liquid in vessel, ft. IC = constant 1 = distance of point on blade from center of rotation, om. L = length of blade, om. Lo = length of paddles, ft. 71 = rotational speed, revolutions per seo. N = rotational speed, revolutions per min. P = horsepower Re = Reynolds numober = nd2p/H, no dimensions t = temperature, C . T = torque, gram-om. V = relative velocity, cm./sec. w = width of paddle blade, ft. W = width of paddle blade, cm. y = clearance, cm. z = absolute viscosity, Ib./(sec.) (ft.) CY, p = constants Y = densitv. lb./cu. ft. . = density; grams/cc. 0 = stirrer-blade pitch angle, degrees = absolute viscosity, grams/(sec.) (cm.) p
TABLEVIII. CALCULATION OF POWERREQUIREMENTS FOR
A
TURBINEIMPELLER IN WATER
[Design details: 48 in.-diam 45' 10-blade metal turbine'rotated in a 12 ft.-diam. tank with 4 vertical dee Gith d a t e r a t 20' C. Clearance not known but un?mportant a4 the high Reynolds baffles. Tank filled 6 numbers used. Direction of)rotation corresponded to the reverse rotation in this investigation. A = 1150 aq. in.; A. 7 1150 sq. in.; A / A 8 l . p O ; d/db = 12/4 = 3.00. This turbine corresponds in size t o t h a t required for a 12 ft.-dism. tank of standard design. Re = n d a p / p = (12)1 X 62.4 X n/0.000672 = 1.34 X 107% dl (for d i n om.) = 6.57 X 1012; T (for T i n gram-am.) = cdspnz = 6.57 X 10%2c; P (in h. p.) = (8.25 X 10-7n)(6.57 X 1012)n% = 5.42 X 1O'nb.I
ft.
n 0.500 0.667 0.833 1.00 1.17 1.33 1.50 1.67 From From From d From a b c
Re 6.70 X 106 8.95 1.12 X 107 1.34 1.57 1.78 2.01 2.24 Figure Figure Figure Figure
-
Correction Factors Revefsed Half Bafflesb rotationc fulld 3.2 1.04 0.90 3.4 1.05 0.90 3.5 1.06 0.90 3.5 1.06 0.90 3.6 1.07 0.90 0.90 3.7 1.07 3.7 1.08 0.90 3.8 1.09 0.90
Standard= Design c 2 . 8 X 10-7 2.7 2.6 2.5 2.5
2.5 2.4 2.4
Corrected C
8.39 X 10-7 8.66 8.68 8.35 8.67 8.91 8.62 8.95
Horsepower Calcu- Meas- % lated ured 0.6 1.0 1.4 1.9 2.7 3.3 4.5 5.7 7.5 9.1 11.4 13.0 15.8 17.5 22.6 24.0 Average
devirtion
26 18 21 18 12 10 6 16
3. 11 ourve D. 11: curve C. 11, curve E.
TABLEIX. CALCULATION OF POWERREQUIREMENTS FOR A TURBINEIMPELLER IN WATER IS SMALLER THANSTANDARD DESIGN WHENTHE IMPELLER
-
[Design details: All conditions, exce t for blade size, similar to.those in Table VIII. 30 inch-diam 45' 10A = 690 sq. in ' Aa = 1150 sq. in.; A / A e = 69$/115b blade metal turbine used. Effective glade area 0.60; d / d b = 12 X 12/30 = 4.8; Re = 1.34 X lid7n; P = 5.42 X 1O6n3c.1 Horsepower Correotion Factors Calcu- Meas- % devirStandard= Reversed Half Blade Corrected lated ured tion Re Design c Bafflesb rotation6 fulld size6 C n 3.4 1.05 0 . 9 0 0 . 2 1 1.82 x 10-7 0.667 8.90 x 106 2.7 x 10-7 1.34 X 10' 2 . 6 3.6 1.06 0 . 9 0 0 . 2 1 1.88 1.00 1.78 2.5 3.7 1.07 0.90 0.21 1 . 8 7 1.33 2.4 3.8 1.08 0 . 9 0 0 . 2 1 1.86 2.23 1.67 2.67 2.3 3.9 108 0.90 0.21 1.83 2.00 3.11 2.3 3.9 1.07 0.90 0 . 2 1 1.86 2.33 3.34 2.3 4.0 1.09 0.90 0.21 1.88 2.50 Average 16 a From Figure 3. b From Figure 11 curve D. C From Figure 11: curve C. d From Figure 11, curve E. 0 315 Figure a At Re c 8 9 X 106, c = 8.5 X 1 0 - 8 for A / A . = 0 6 0 - correction = 8.5 X 10-8 2 7 X 10-7 10). At Re ='8.9 X 106, c = 3.0 X lo-* for d / d b 4k;'correction = 3.0 X !O-S/2.7 X lo-! = 0:111 [Figure 0.111)/2 0.21; this remains substantially oonstant, independent of eR 10). Average correction = (0.315 over the range involved in this table.
+
--
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INDUSTRIAL AND ENGINEERING CHEMISTRY
208
Vel. 34, No. 2
5 4
3
Z
0
i=2
u
2 1.5 E
E,
88
0.7 0.6
-
2 4 68IO32 4 68IO42 468105 2 468106 2 4 6810’ 2 4 6 8 Re n d‘p P
FIGURE11. CORRECTION FACTORS FOR SINGLEVARIATIONS IN DESIGNFROM STANDARD A. B.
THE
F. Vessel three fourths full
60’ turbine Paddle agitator C. Reversed rotation D. Two and four baffles E . Vessel half full
-
0. Low clearance, y / d = 1/12 H . High clearance, y/d ‘/a J. Large impeller, d / d a = 2.35
Acknowledgment The authors are deeply indebted to the New England Tank and Tower Company of Everett, Mass., who made available their data on the power requirements of agitating equipment. Acknowledgment is also due the Newark College of Engineering, which made possible the inauguration of this research.
Literature Cited (1) Badger, W. L., and McCabe, W. L., p. 517, “Elements of Chemical Engineering”, New York, McGraw-Hill Book Co., 1936. (2) Badger, W. L., Wood, J. C., and Whittemore, E. R., Chem. dl. Met. Eng., 27, 1176 (1922). (3) Bissell, E.S., IND. ENG.CHEM., 30, 493 (1938). (4) Brothman, A., Chem. & Met. Eng., 46, 263 (1930). (5) Buohe, W., 8. Ver. deut. Ing., 81, 1065 (1937).
(6) Hixson, A. W., and Baum, S. J., IND.ENG. CHEW, 33, 478 (1941). (7) Hixson, A. W., and Luedeke, V. D., Ibid.,29, 927 (1937;. (8) Hixson, A. W., and Tenney, A. H., Trans. Am. Inst. Chem. Engrs., 31, 113 (1935). (9) Hixson, A. R’., and Wilkens, G. A., IN^. ENG.C H m f . , 25, 1196 (1933). (10) MacLean, G., and Lyons, E. J., Ibid., 30, 489 (1938). (11) New England Tank and Tower Co., Catalogs, 380 (1937), 410 (1940). (12) Valentine, K. S., and MacLean, G., Sect. 14 of Perry’e Chemical Engineers’ Handbook, New York, McGraw-Hill Book Co., 1934. (13) Walker, W. H., Lewis, W. K., MoAdams, W. H., and Gilliland, E. R., “Principles of Chemical Engineering”, p. 298, New York, McGraw-Hill Book Co., 1937. (14) White, A. M., Brenner, E., Phillips, G. A., and Morrison, M. S., Trans. A m . I n s t . Chem. Engrs., 30, 570 (1934); White, A. M., and Brenner, E., Ibid.,30,585 (1934). (15) White, A. M., and Sumerford, €3. D., Chem. & Met. Ew., 43, 370 (1936).
TABLEX. CALCULATION OF POWER REQUIREMENTS FOR TURBINE IMPELLERS IN VISCOUS LIQUIDS [Results of ru,n made by the New Englafld Tank and Tower Co. Design details: Impeller, a 42 in.-diam. turbine with 10 blades; effective blade area, A = 977 sq in.; reverse rotation. Clearance not exact1 known b u t corresponds closest t o y / d = 1/12 used in present investigation. Ves’sel a cylindrical t a n k with a truncated cone%ottom. four vertical bafflesin upper 8ection depth of truncated section 16 in.; Iaigest diam. of tank, 96 in.: narrowest diam. (at bbttom), 50 in. Fluid, heavy roFd oil. ‘viscoaity 6200 centipoise, sp. gr. approx. 1.0; I100 gal. used in test. T o use the curves shown In the present inveitigation the standard tank diameter corresponding to the impeller diameter must be used in evaluating the plotted groups. d (for 42 i&diam. turbine) = (42/12) 3 = 10.5 ft.. A . (for a 10.5-ft. tank) = 882 sq. in. Then A / A , = 977/882 = 1.10: R e n d * p / p = n(10.5)* X 62.4/(0.000672 X 6200) = 1650n: d’(for d i n em.) = 3.36 X 101%; T 3.36 X I 10%2o; P 2.77X
-
n
0.333 0.667 1.00
-
k
-
-
10BnSc.I
Re
Standard’ Design c
Clearanceb
550
1.9 X 10-6
1.2 1.3 1.4
1100 1650
1.3
1.1
1.33
2200
9.7
1.67
2750
2.00
3300
9.0 8.7
x
10-7
1.5 1.6 1.7
Correction Factor8 Reversed rotatione Aread
0.82 0.86 0.88 0.90 0.90 0.91
Corrected C
1.10
1.10
2 . 0 6 X 10-6 1.60
1.10 1.10 1.10 1.10
1.49 1.44 1.43 1.48
Calculated
b
0.2
1.3 4.1 9.4 18.4 32.8
From Figure 3. From Figure 11 curve G. c From Figure 11’ curve C . d No curve availLble for size correction of impeller in laminar flow region: assume correction tion required for d/db. a
-
Horsepower Aweas- yo ured 0.8 1.6
3.8 9.1
..
19
8 3
17.6
5
30
9 Average
-
:;:9
A / A a = 1.10. No correc3
