OF SOME GEOMETRIC PARAMETERS OF IMPELLER POWER
AN EXAMINATION
ROBERT
L.
BATES,
PHILIP
L.
FONDY,
AND
ROBERT
R.
CORPSTEIN
Chemineer, Inc., Dayton, Ohio
Downloaded via UNIV OF WOLLONGONG on June 18, 2018 at 19:51:13 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
The simplified equation for impeller power is well known and power coefficients have been published for most impellers. But many of the parameters of impeller and system geometry which appear in the gener-
alized form of the equation are not well established. This paper is a study of the effect of some of the shape factors on impeller power. Including consideration of both impeller and vessel geometry, data are presented on impeller blade width, number of blades, blade angle, proximity of impellers to system boundaries, shrouding, spacing of multiple impellers, and extent of baffling. Both radial and axial flow turbine impellers are treated. New basic Power number curves are presented for the full Reynolds number range for a variety of turbine impellers. Unwin (15) delivered a paper before the Royal in London in 1880, there have been over 100 publications on the subject of impeller power. This would seem to indicate a subject so well explored as to allow little But this is far from the case. room for further fruitful study. In the early studies the existence of a transition flow range and the effect of flow pattern on power were not fully appreciated. The tendency then, in attempting to write a “power equation,” was to select average exponents for the various variables when actually operation under several flow conditions was represented. Experimental facilities were often crude and even in some relatively recent papers the power data are not reliable. The trend in recent years toward use of the simplified form of the power number for all correlations has resulted in much apparent conflict of data. The initial purpose of this paper is to re-introduce the expanded form of the power equation, to establish perspective for this and future discussion of impeller power. Then, some of the terms representing impeller and system geometry are examined and new data on their effect on power presented.
Singe Society
Background of Rower Theory White and coworkers (77General Power Equation. 19) were the first to point out the possibility and advantage of correlating impeller power by dimensional analysis. To them goes credit for originating the drag coefficient group now known as the Power number. The development of a full generalized form of an equation for correlating power has been well covered by Hixson and Luedeke (5), Johnstone and Thring (6), and Rushton, Costich, and Everett (13). A relationship in its full form, using impeller diameter, D, as the referenced length, is:
©'© ©‘©‘
«>
The dependent variable in Equation 1 characterizes the basic flow pattern, as proposed by White and Brenner (77), and is called the Power number, JVP. The first independent dimensionless group is readily recognized as the impeller Reynolds number, iVRe. The second group is the Froude 310
l&EC
PROCESS DESIGN
AND
DEVELOPMENT
number, NFr, theoretically required to account for the vortex formation in a swirling system. The next seven terms in Equation 1 define the effect of system and impeller geometry. The last term is not a linear dimension ratio but is required to account for a change in number of impeller blades, referenced to some standard design basis. To be fully inclusive, the equation should be expanded to include: bafffe width and number of baffles; spacing of multiple impellers; and offcenter impeller positioning. Simplification of Power Equation. The last seven terms of Equation 1 and the three just mentioned are parameters of geometry. If geometric similarity is stipulated and a nonswirling system employed, the equation simplifies to JVp
=
K(NR')‘
(2)
The justification for this simplification is twofold. First, it makes for easier evaluation of the more critical factors affecting power since, admittedly, the effect of many of the shape factors can be minor. And it recognizes that geometric similitude is easily maintained in scale-up work—and is usually desirable. For illustrative purposes and for many applications, the simplified version is acceptable. But the very ease with which it can be used has led to widespread use for correlating all power data, irrespective of geometry. The result has been some confusion and several useless correlations. There are many ways of treating the shape factors in correlating power. The full general equation could be used but is obviously cumbersome. The conventional plot of log 7 vs. log A7Ee is undoubtedly the best way to present the basic power behavior of an impeller, but its use would seem to be best restricted to representing a stipulated and essentially standardized set of conditions for impeller and system geometry. Deviations from the standard can then be treated individually or collectively in best adapted to the particular variables. Forethe manner the results of this study, we find that some of the glimpsing parameters may be analyzed individually and that others are interrelated and must be grouped. Thus, it may be anticipated that a simple exponential representation as indicated in Equation 1 is not always feasible or especially helpful for many of the geometric factors.
Experimental of this paper is derived from a broad study of all types of impellers—radial, axial, and tangential—in the range of flow from laminar to fully turbulent. Since the majority of geometric factors are prevalent in the turbulent range and, in practice, with turbine-type impellers, the data presented here (with the exception of Figure 1) are for these conditions. Table I presents the range of variables used in this work. In the portion of this study where it was desired to maintain geometric similitude, the following “standard conditions” were used: D/T l/z,C/T Vi, Z/T 1, 4, tt>t> T/12.
The
content
=
=
=
=
=
Table I.
Range of Variables Used in This Work
6, 10, 12, 15, and 24 inches Turbine diameters. 3, 4, 5, 6, 7, 8, and 10 inches Turbine styles. Flat six-blade disk style; flat, curved, and 45° pitched six-blade open styles; ° four-blade open styles with blade angles 25 to 90
Vessel diameters.
w/D range.
0.062 to 0.37 range. 0.25 to 0.50 1 through 12 ru, range. wh/T range. 7 to 15% ribWt/T range. 0.1 to 1 Newtonian fluids used. Water, corn Viscosity range. 1 to 120,000 cp. Vr, range. 10_I to 10s
D/T
sirup
The variable-speed drive and dynamometer assembly used in of the work was the Model ELB assembly manufactured by Chemineer, Inc., and described by Bates (7). A larger version of this system was used for runs in the 24-inch diameter vessel. Impeller dimensions were controlled very accurately. This was accomplished here by using investment castings for the most
Figure 1.
standard impellers and machined fabrications for the special styles studied. The projected blade width of the pitched blade impellers was rigorously controlled by machining both blade edges in the plane of rotation. This was necessary to eliminate any possible “edge” effect in the smaller impellers.
Impeller Geometry Figure 1 is the conventional log-log plot of the simplified power equation and is presented here to submit new data on radial discharging impellers and to illustrate the characteristic These data represent over curves for different impeller styles. 1000 points recorded for the various designs under the “standard” conditions in Newtonian fluids. Style. Curve 1 is the correlation from this study for the radial discharging six-blade turbine impeller on which data were originally presented by Rushton, Costich, and Everett (74). Several investigators (3, 70) have since obtained valuesof the Power number lower than the 6.3 reported by Rushton for the turbulent range. The data of this study indicate a value of 4.8 for four T/12 baffles and 5.0 for four T/10 baffles.Noteworthy is the finding that a difference in power requirement exists between the disk style of construction and the flatblade turbine in the turbulent range. Curve 2 is this open flat-blade style with a full blade originating at the hub. Though it has a longer blade than the disk style, it consumes approximately 25% less power. Since this study was concerned only with impeller power consumption, it has not been established whether the vaned-disk construction yields a higher performance for its increased power requirement. For the same “standard” system conditions, full curves are also shown for flat, curved, and 45° pitched-blade open style six-blade turbines with w/D of 1/i.
Power number-Reynolds number correlation in Newtonian fluids for various turbine impeller designs VOL.
2
NO. 4
OCTOBER
1963
311
Thus the effect of blade width and number of blades for these designs is seen to be interrelated. It is not possible to establish the independent effect of blade width and number by the simple (w/D)g and (n-¡/n/)' from Equation 1. The easiest approach involves preparing a plot similar to Figure 2 for the style of impeller being considered and obtaining power correction factors directly. ° Impeller Pitch. The universal use of the 45 blade angle for pitched-blade turbines has resulted in a complete lack of data on the effect of blade angle on power. The two frequently cited sources, Hixson and Baum (4) and Van de Vusse (16), are for unbafffed systems and thus not helpful for the usual application. Since turbine impellers have a constant blade angle, as contrasted with the helicoidal design of propellers, the term “pitch” has no real significance. Thus, this correlation of power with pitch uses a function of blade angle rather than the (p/D) term of Equation 1. Figure 3 is the correlation for four-blade open-style pitched turbines operating in a “standard” system. Projected blade width in the elevation (w sine ) was held constant for each series of runs. The abscissa term F6/F90o is simply the ratio of power in a given condition of pitch to the power consumption of a radial discharging style with vertical blades. Over a blade angle range of 25° to 90° then, we may substitute—for fourblade turbines—for the (p/D)’ term the expression (sine #)2·5. Shrouded Impellers. A radical discharging turbine is frequently equipped with a partial or full plate on the top and/or bottom to control the suction. The few writers who mention this style of construction used impellers similar to a centrifugal pump impeller and the separate influence of the shroud is not indicated. Lee, Finch, and Wooledge (7) used a shrouded disk turbine in their high viscosity work, and their data for the laminar and transition range indicate an increase in power of as much as 50%. From this study, data on the effect of the two most common shroud modifications are available. The basis is operation in the turbulent flow range, with impeller location and baffle design as stated earlier for “standard” conditions. With the shroud plate fully covering the top of the turbine, the power increase is 30%. With a full bottom shroud, the increase is two
Figure 2.
blades
on
Effect of width and number of turbine
power
Blade Width and Number of Blades. In using the simplified relationship where power is proportional to D3 in the laminar and D3 in the turbulent range it is often assumed that w is equivalent to a D term and that power is directly proportional to the blade width. This is not true in the laminar and transition range but has been generally thought to be true for the turbulent case. The fact that the exponents of all linear dimension terms will total the exponent of the diameter term merely confirms the cumulative effect of impeller geometry. The exponent of the D term must always be recognized as of variable composition. In the fundamental study of O’Connell and Mack (12), where open-style radial discharging turbines were used, blade width and number of blades were found to be interdependent variables. For the six-blade design they found power to vary as (w/D)1·03. Their four-blade design had a blade width eflect of (wD)1·13. Figure 2 is the correlation from this work for both the fourand six-blade open-style designs again operating in the “standard” system. The range of 0.067 to 0.37 w/D ratios more than spans those commercially used. For the four-blade design, the exponent of the w/D ratio was found to be 1.25 but the six-blade was 1.0.
Figure 3. 312
I
&
E
C
Effect of turbine blade angle
PROCESS
DESIGN
AND
on
power
DEVELOPMENT
47%. System Geometry
The environmental effect of fluid properties is generally well appreciated and always included in evaluation of a given impeller. But the external factors of geometry relating the impeller to the system boundaries and influencing flow pattern have received remarkably scant attention. D/T and Baffles. An investigation to confirm the published statement that the ratio of impeller to tank diameter (D/ T) has no effect on power revealed that there is an effect—at least with the flat open-style six-blade turbine—and that this variable is interrelated with the extent of antiswirl baffling present. The phrases “100% baffling” or “full baffling” are commonly used in both academic and industrial parlance, but are variously defined in terms of the number and width of baffle plates. These are, of course, assumed to be conditions which give approximately maximum power consumption. Four flat baffles are most frequently used. Many fundamental studies have used a baffle width of T/10, but industrial practice is almost universally /12. The precise effect on power of variation in number of baffles and baffle widths has been presented in two contemporary works, although the findings are not in agreement. Bissell et al. (2) tabulated, without supporting data, the
per cent power based on four T/12 baffles and show an increase in power above four baffles and above 7712 width. Mack and Kroll (8) found a limiting condition of number and width of baffle, above which no increase in power occurred. The recent work of Nagata and associates (77) shows that the power for a given number of baffles reaches a maximum and then decreases somewhat as width increases. An approximation of their results gives the relation
02
0.1
BAFFLE
Figure 4.
0.3
RATIO
-
V"
Effect of baffling and D/T on power
3AFFLE
Figure 5.
RATIO
Composite
curve
from Figure 4
for maximum power consumption. This result, however, was based on a study involving a two-blade impeller in only one vessel diameter. Figure 4 is a summary of data for four D/T ratios. Each point was computed from r.p.m.-horsepower correlations (approximately 1000 recorded data points) of six flat openstyle six-blade turbines in three tank diameters using nb baffles 1 through 12 in wb/T ratios from 7 to 15%. It is apparent that a variable power effect results from changes in D/ T ratio and opposed baffle area. Figure 5, combining the curves from Figure 4, shows the combined effect of D/T ratio and baffling. Data points are omitted for clarity. At nbwb/T values less than 0.20, measured Power numbers decrease with increasing D/T ratio. Conversely, at nbwb/T values above 0.35, measured Power numbers increase with increasing D/T. The industry standard of four 77'12 baffles gives an nbwb/T value of 0.33. Figure 5 shows that the change in Power number with D/ T ratio is essentially negligible, within 5% for the D/T range studied. The studies of Rushton et al. (74) indicate no effect on power for variation in D/T. The only significant difference between that work and this study is that a disk-style turbine was used. Shape Factors. Practically all power studies have been made in vertical cylindrical vessels. Data on other arrangements which occur in practice are limited. Table II shows results for several common applications for flat open-style sixblade radial discharging turbines with impeller diameter 40% the width or diameter of the tank and bottom clearance at one impeller diameter. The factor shown is the ratio of impeller power drawn in the specific geometry to that for “standard” conditions. Even though equivalent power consumption is apparent for several situations, there is a drastic difference in the nature of the flow pattern. Choice of the style of installation would be determined by the particular process requirement. Spacing. The effect of impeller spacing on power is shown in Figure 6. Spacing as used here is the vertical dimension between the bottom edges of the two turbines; a spacing of 0 indicates complete coincidence of the two impellers. In the ratio Pi/Pi, the reference power, Pi, is a flat open-style sixblade turbine in all cases. The 45° pitched-blade turbine then falls lowest and the combination of the two styles is intermediate. Both styles have w/D V s· Within a spacing of four impeller diameters, dual pitched-blade turbines do not yet equal twice the power consumption of a single. The combination of two types, the pitched above the flat, reaches a level of the sum of the two at about one diameter spacing. But two flat-blade turbines actually develop a total power almost 25% greater than the sum of the two when the spacing is less than one diameter. Proximity to Tank Bottom. This study shows that the space beneath a turbine impeller has a definite effect on power. These data are in disagreement with those of Mack and Kroll (8), who noted no change in power for two-blade turbines over a range of C/D values from 0.35 to 2.5. =
P2/p
-
RATIO
POWER
Figure 6.
Effect of dual turbine spacing on power
VOL.
2
NO. 4
OCTOBER
1963
313
Table II.
Vessel Shape Power Factors for Six-Blade Open-Style Turbine
w/D
=
1/8 Vertical
Impeller
Installation
D/T
Horizontal cylindrical tank, 5:1
0.4
Location
Number
Center mounted
0.4
Center mounted T/4 eccentric mounted Center mounted
To indicate clearly the typical behavior of the three styles of turbines, Figure 7 shows the Power number for the ordinate rather than a power factor. For direct use as a geometric factor in Equation 1, the ratio of the NF value for a specific condition to the NF value at “standard” conditions—i.e., C/T V3—can be For the disk disk, there is a creased. The =
used.
Location
Factor
D/4 distance, 180° longitudinal
1.0 1.0
None
(2) Square tank
ßß5
/10
axis
None None 90 °, wall center
(4) Y/IO (2) /10
T
—
w
=
wb
=
Z
=
180°, wall center
0.75 1.0 1.0 1.0
tank diameter impeller blade width baffle width liquid depth
Greek Letters angle of impeller blade from horizontal =
turbine, since the suction is partitioned by the marked reduction in power as clearance is deflat open-style six-blade turbine (w/D Vs) displays a variable effect at different clearances, but in general a slightly higher power level at lower values of C/D. Increasing the proximity of a 45° pitched open-style sixblade turbine (w/D Vs), as expected, increases power consumption. The data of Miller and Mann (9) note a reduction in power rather than an increase, but their data were taken in an unbaffled system and thus are not comparable. =
=
µ
=
viscosity density
Subscripts 1 condition condition 2 =
1
=
2
Exponents
=
a, b, c, etc.
=
Figure 7.
Effect of turbine proximity
on
power
=
D
=
gc
impeller distance off tank bottom, measured from underside of impeller impeller diameter gravitational constant or conversion factor
=
K
=
/
=
blade length number of impeller blades number of baffles impeller speed, r.p.m. Froude number Power number Reynolds number blade pitch power impeller spacing
=
IV
=
AV
=
Nf
=
vVRe
=
p P
=
S
314
Scale-up Methods in Chemical Engineering,” McGraw-Hill, New York, 1957. (7) Lee, R. E., Finch, C. R., Wooledge, J. D.. Ind. Eng. Chem.. 49, 1849 (1957). (8) Mack, D. E., Kroll, A. E., Chem. Eng. Progr. 44,189 (1948). (9) Miller, S. A., Mann, C. A., Trans. A.I.Ch.E. 40, 709 (1944). (10) Nagata, S., Yokoyama, T., Mem. Fac. Eng., Kyoto Univ.. (Japan) 17, 253 (1955). (11) Nagata, S., Yokoyama, T., Maeda, H., Ibid., 18, 13 (1956). (12) O’Connell, F. D., Mack, D. E., Chem. Eng. Progr. 46, 358 (1950). (13) Rushton, J. H., Costich, E. W., Everett. H J.. Ibid., 46,, 395 (1950). (14) Ibid., p. 467. (15) Unwin, W. C., Proc. Roy. Soc. (London) A31, 54 (1880). (16) Van de Vusse, J. G., Chem. Eng, Set. 4, 178, 209 (1955). (17) White, A. M., Brenner, E., Trans. A.I.Ch.E. 30, 585 (1934). (18) White, A. M., Brenner, E.. Phillips, G. A., Morrison, M. S., Ibid., 30, 570 (1934). (19) White, A. M., Somerford, S. D., Chem. Met. Eng. 43, 370 (1936).
A.I.Ch.E. Meeting, Chicago,
111.,
December 1962.
constant
=
nb
Bates, R. L., Ind. Eng. Chem. 51, 1245 (1959). Bissell, E. S., Hesse, H. C., Everett, H. J., Rushton, J. H., Chem. Eng. Progr. 43, 649 (1947). (3) Calderbank, P. H., Trans. Inst. Chem. Engrs. 36, 443 (1958). (4) Hixson, A. W., Baum, S. J., Ind. Eng. Chem. 34, 194 (1942). (5) Hixson, A. W., Luedeke, V. C., Ibid., 29, 927 (1937). (6) Johnstone, R. E., Thring, M. W., “Pilot Plants, Models and
(1) (2)
Received for review December 31, 1962 Accepted June 10, 1963·
Nomenclature C
Literature Cited
= =
I
&
E
C
PROCESS
DESIGN
Correction
THE KINETICS
OF NICKEL CARBONYL
FORMATION
In this article by W. M. Goldberger and D. F. Othmer [Ind. Eng. Chem. Process Design and Develop. 2, 202
(1963)], on page 209, reference 8 should read: Othmer, D. F., Luley, A. H., Ind. Eng. Chem. 38, 408 (1946). AND
DEVELOPMENT
