FIG. 20. Power per unit volume for equal process result. From Bates (B3).
volume being the ordinate and batch size the abscissa. The extent to which one should extrapolate this plot is, for one thing, a function of the accuracy of the slope, which is dependent upon the increment of batch volume obtainable.
Naturally, t o o , the confidence in extrapolation decreases as the slope departs from zero which represents constant power per unit volume. In Fig. 20 curves Β and B' are typical deviations. Curves C and C would be considered wide variation from constant horsepower/volume and would indicate a pro-cess demanding more detailed study.
Power per unit volume scale-up has been discussed in the literature ; Buche (B12) and Hirsekorn and Miller (H3) on solids suspension, Treybal (T2, T3) and Barker and Treybal (Bl) on liquid-liquid mass transfer and gas-liquid studies in references (C8, H9, K 3 , K4) are each of particular interest.
b. Dimensionless group correlations. Dimensionless groups are ratios of like quantities (lengths, forces, velocities, etc.) and permit a generalized characterization of the interaction of the physical variables in a system. A n important result of the dimensional method is the ability to study the vari-ables in a particular system under ideal conditions with model fluids^ then predict the results with any other fluids by substituting the magnitude of the physical quantities. A n example is the NP vs. JVRe curve in Fig. 7 which can be easily developed with simple fluids such as water and corn syrup. The re-sulting correlation can then be used t o estimate power requirements with toxic, corrosive, or explosive fluids which would be inconvenient for direct one vessel size with the results scaled directly on a power basis without any appreciation of the underlying mechanism of the mixing operation. If the mechanism is not well understood the only safe approach involves use of two or more geometrically similar vessel sizes. A log-log plot (Fig. 20) of the type shown by Bates (B3) is then used to relate the two variables, power per unit
study. Another equally important result of a dimensionless group correlation is the inherent features for use in scale-up, although errors in treatment of data are c o m m o n .
Consider the problem of simple blending of miscible liquids. It is common-ly thought that data should be correlated as mixing time vs. Reynolds number for a particular geometric configuration, and by holding geometric similarity in scale-up, equal Reynolds number will yield equal mixing time. This is in-correct as the mixing time has not been properly expressed. Another inin-correct analysis would presume that equal power per unit volume would yield equal mixing time in scale-up. A more sophisticated but still incorrect approach would insist on complete geometric and dynamic similarity in scale-up—but similarity does not predict equality of process result. The correct approach (B8) is to express the process result, mixing time, in dimensionless form, Nt9
and correlate this group with the appropriate dimensionless groups of signifi-cant variables, Reynolds number, F r o u d e number, etc. Scale-up of this re-sulting correlation follows directly and is covered fully in Chapter 4. It is of interest to note that for equal mixing times the power per unit volume in-creases in scale-up.
The most significant result of the above example is the importance of ex-pressing the process result, mixing time, heat and mass transfer coefficients, etc. in dimensionless form. Correlations involving heat and mass transfer are covered in Chapters 5 and 6 (Vol. II). The existence of these tools is invaluable but certain precautions are necessary. Serious error can result, for example, if all variables are not included, and many times a significant variable is not intui-tively apparent. Geometric similarity is not always easily maintained. F o r example, it is often not convenient to scale down particle size in a solid-liquid study. This can lead to serious error in scale-up. Further, even with geometric similarity the particle surface area per unit volume decreases with scale-up, an important consideration for dissolving operations. F o r further reading, Johnstone and Thring (J2) present a systematic development of the concepts of modeling and techniques of scale-up.
VIII. Agitator Drives for Experimental Use
The extent of data obtainable from an experimental unit for use in scale-up is often limited by an inadequate agitator drive. M a n y bench and pilot plant units are not even variable in speed which is one of the primary features of a well-designed experimental unit.
A . DRIVE SELECTION
Today, many small, variable-speed drives a b o u n d — b u t only a few have all of the features desirable for experimental use. The essential requirements are wide speed range, power characteristic of at least constant torque, good speed
3. Impeller Characteristics and Power 169 regulation under varying load, availability with an explosion proof motor,
and compactness and ease in mounting. These spécifications are for a drive intended for multipurpose small-scale service and, naturally, all are not mandatory for a given application.
7. Speed Range
The breadth of the speed range is not dictated by the need to vary power investment; with Ρ oc Ν3, a small speed variation will be needed for a given impeller. Rather, ample range must exist to allow a shift in impeller size and type. Typically, a general purpose \ h p . bench-scale drive should be adjust-able over a 10:1 range, e.g., 110 to 1100 r.p.m. Pilot plant units usually pre-suppose operation with only one impeller style or at least a somewhat limited flow/shear variation and thus a speed range of 3:1 to 4:1 is c o m m o n . Because of the sensitivity of power and performance to small speed changes, a con-tinuously variable control is needed. Obviously, multispeed m o t o r s have no application in this work other than possibly extending the range of a variable speed mechanism.
2. Torque Characteristic
The torque design of variable speed drives used for experimental work must be judged on a different basis than with single purpose agitators. In the latter, advantage can be taken of the aforementioned relation of power to speed to utilize a low-cost variable torque design. But in experimental studies the specific problem and selected impeller system may impose any load at any speed in the range. Thus, select a drive on the basis of its capability in the particular speed area anticipated immediately, but do not settle for less t h a n a constant torque design.
3. Speed Regulation
Speed regulation here refers to the behavior of the drive under varying conditions of load. If for no other reason than to insure valid and consistent data, it is important that the agitator speed not fluctuate with the changes in batch fluid properties which often occur during a run.
4. Classification
In the light of the preceding requirements, some comment on the commer-cially available drives should be m a d e . In the approximate order of increasing costs :
a. Small stirrers in a wide variety are found in laboratories but most are useless for investigative purposes.
b. Brush shifting or variable voltage wound-rotor repulsion a.c. motors are least expensive but p o o r load-speed regulation renders them unsuitable for experimental work.
c. Eddy current clutch drives have good regulation ( ± 3 % ) , a broad speed range (1650/165 r.p.m.), and a constant torque characteristic. Unfortunately, they are available only with an open enclosure.
d. Variable pitch pulley belt drives have good regulation and torque features but are somewhat bulky and cumbersome for a bench-scale setup.
Available in both " p a c k a g e " styles and in the form of components, these drives are widely used in pilot plant designs in the large fractional and small integral horsepower drives.
e. Metallic traction drives satisfy all of the requirements and have particu-larly good regulation. Allowance must be made for a transmission loss of approximately 25 %. Ideal for bench-scale units, the cost and bulk become excessive when considered for larger pilot plant drives.
/. Direct current motor drives are widely used, employing shunt or com-p o u n d wound motors with rectifier and transformer control from an alternat-ing current source. Speed regulation varies from less than 1 % to as high as 3 0 % , depending on the type of winding and quality of control. Control is a separate component and must be located outside hazardous areas or entail the considerable expense of a special enclosure.
B. POWER MEASUREMENT
F o r most low viscosity fluids in equipment of conventional geometry, power can be calculated directly by using a basic impeller rating and correct-ing for liquid density. But a more exact method of derivcorrect-ing power data is necessary if the liquid is non-Newtonian, if a gas phase occurs, or if unusual geometry or flow pattern conditions exist.
1. Electrical Methods
If possible, electrical methods of obtaining power data are to be avoided.
Operation may be below the no-load current rating of the motor. Also, losses in transmission components and seals may be included. If wattmeter or am-meter readings are necessary, the former are preferred and it would be advis-able to obtain a specific m o t o r performance curve from the manufacturer. In the absence of a n individual m o t o r curve, the typical relations shown in Fig.
21 will be helpful. A brief discussion of the accuracy of various electrical methods is included in reference (A3).
2. Dynamometers'
A paper by Nagata and Y o k o y a m a (N2) deals at length with the various methods of measuring power of agitator assemblies, and this reference should be consulted for an evaluation of relative merit.
a. Vessel reaction. The popular design of early workers was a pivoted counterbalanced vessel. White et al. (W3) and Hixson a n d Luedeke (H5) used this type and present details of construction. The last reference shows a cross
3. Impeller Characteristics and Power
section of the pivot bearing arrangement. The disadvantage of this dynamo-meter arrangement is the need t o keep the t a n k free of external connections such as fluid lines and thermocouple leads. Hippler (H2) in a brief study simi-lar t o that of Hirsekorn a n d Miller (H3), b u t using a vessel-reaction setup, concluded that their results a r e low because of use of a " t o r q u e t a b l e . "
Standart (S5), in a mathematical analysis, shows that t h e vessel-reaction arrangement is suitable only for t h e special case where the agitator shaft is vertical a n d coincident with the turntable axis.
b. Drive reaction. A more direct approach t o the problem is t o permit the drive t o rotate o n its axis and then register the torque via a m o m e n t a r m and scale. Figures 22 and 23 show schematically two setups of this type, using a commercial ball thrust bearing. I n either, it is necessary t o establish t h e bearing friction constant with the impeller in t h e air. This can b e done by
FIG. 22. Drive reaction dynamometer, balance scale (schematic).
1 6 0 . —j —ι — ι — ι — ι — ι 1—1—| 1—ι 1—ι 1 1 6 0 ι—ι—ι—ι—ι 1 1—ι—ι—ι 1 ι ι — Τ
-Ι
1 4 0 Γ " £ 1 4 0i
1 cjr*_ < a
3 . 2 0 S 1 2 0 -Χ—
ο g_ 2 C,
3 too ^ , —
I
ιοο — Ζ.J ^ J 7
2 8 0 j 8 0
S Λ ' 3
5 4 0 - ^ * 4 0 J£
s -vl ι ^
e o
.
2 0_ ^
1 1 1 1 1 1 1 1 1 1 1 1 1 ri * 1 1 I I I I I 1 1 11111—
0 2 0 4 0 6 0 8 0 1 0 0 120 140 0 2 0 4 0 6 0 8 0 KK> 120 MO PER CENT RATED LOAD PER CENT RATED LOAD
FIG. 21. Motor performance curves, (a) 13 hp. (b) 57£ hp. (c) 10100 hp. (d) 5 -100 hp. (Courtesy of Westinghouse Electric Corporation.)
172 Robert L. Bates, Philip L. Fondy, and John G. Fenic
FIG. 23. Drive reaction dynamometer, spring scale (schematic).
using the scale to apply a gradual load on the drive arm. Figure 24 illustrates a bench-scale setup of this type. The stand is designed to handle vessels as large as 16 gal, and is equipped with adjustable legs and a level indicator. Details of construction of an extremely sensitive and friction-free design in which the m o t o r is supported in air bearings is given in reference (CI).
c. Deflection. Deflection or torsion dynamometers have the advantage of permitting rigid mounting of both vessel and drive but do usually complicate agitator shaft support. M a n y styles of construction have evolved for measur-ing deflection, but all the devices are based on the principle of registermeasur-ing the displacement at a joint in the agitator shaft. Spring types for small units have been described by Bungay (B13) and Pocock (P2). Detailed construction of a unit suitable for pilot plant and field measurement is given by Uhl ( U I ) .
d. Differential gear. Differential gear types have been widely used—for one, they can be readily derived from automotive parts—in large-scale studies, but the friction of the gear train introduces a serious error when the torque readings are small.
e. Strain gage. Strain gage devices, a form of the deflection type, are manu-factured by several companies such as Baldwin-Lima-Hamilton, Crescent Engineering and Research Co., and Metron Instrument. They are complete instruments and easily adapted to this work but are quite expensive.
IX. Future Needs
Power correlations are now available to cover most situations of practical interest. The major area of study should involve improvement in process re-sults and power efficiency by improved impeller design and application. Some long-term studies are just beginning which will consider the fundamental nature of turbulence in agitated vessels. These studies will certainly shed light o n the mechanism of the mixing process and are to be encouraged. Continu-ing improvements are also encouraged in all phases of agitatContinu-ing non-Newtonian fluids, gas-liquid and liquid-liquid contacting operations, and
3. Impeller Characteristics and Power
FIG. 2 4 . Drive reaction dynamometer with bench-scale agitator. (Courtesy of Chemineer, Inc.)
mixing in continuous flow processes. Few studies can be found which com-pare process performance and economy of agitated vessel systems with other classes of process equipment. F o r example, many contacting operations formerly conducted in packed towers have been handled with low power requirements and high economy in multistage agitated columns.
Very little work has been done in determining which impellers are most effective in the various classes of agitation problems (blending, gas dispersion, solids suspension, etc.). Establishment of the design features which cause a certain type of impeller to be most effective in a given area could lead to more efficient designs.
174 Robert L. Bates, Philip L. Fondy, and John G. Fenic
I n brief, w h a t is most needed in agitation research at the present time is n o t extensive correlations of power c o n s u m p t i o n of inefficient mixing devices b u t intensive study directed t o w a r d design a n d selection of impellers which will utilize t h e power invested m o r e effectively.
List of Symbols A cross-sectional area
C impeller distance off tank bottom, measured from lowest side of impeller CD drag coefficient [Eq. (10)]
Δ C D/(T- D), wall proximity factor D impeller diameter
De D — wL9 effective agitator diameter D'e equivalent agitator diameter [Eq. 27)]
dujdr shear rate [Eq. (34)]
L vertical arm length or a characteristic length Le L—wD/2, effective depth of agitator
L'e equivalent vertical arm length [Eq.(27)]
η number of impeller blades
nt number of effective blade edges [Eq. (26)]
ri flow behavior index [Eq. (37)]
nb number of baffles
N'Re modified Reynolds number [Eq. (10) in reference C4]
ρ blade pitch Δρ pressure difference jP power
Pm&x power, maximum attainable Pg power, gassed
3. Impeller Characteristics and Power 175 List of Symbols {Continued)
Vt superficial gas velocity based on cross section of tank w impeller blade width
wb baffle width
wL side-arm blade width [Eq. (27)]
wD cross member width [Eq. (27)]
X, y volume fraction [Eq. (28)] or displacement of propeller agitator (Fig. 16) Ζ liquid depth
GREEK LETTERS
α angle of agitator shaft from vertical β angle of agitator shaft from tank diameter ε eccentricity of shaft
θ angle of impeller blade from horizontal
Φ
Νρ/ΚΝρτΫ]
or angle of curvature of curved-blade turbine μ viscosity, absoluteμα viscosity, apparent μΜ viscosity, average ρ density
τ shear stress [Eq. (34)]
SUBSCRIPTS
g gassed
x, y components of mixture c continuous phase d dispersed phase w water ο organic
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