Impeller Power Numbers in Closed Vessels Alvin W. Nienowl and David Miles Department of Chemical Engineering, University College London, Torrington Place, London, W.C.L.,England
Power numbers are presented for 6-blade
disk turbines, 2-blade flat paddles, a n d
4-blade, 45”-pitch turbines in 6-in. a n d 12-in. diam closed vessels for Reynolds numbers
>
2 x lo4. The effect of impeller size and clearance above the base i s reported disk turbines are extremely sensitive t o changes in carrier disk thickness. For disk turbines, power numbers for
in detail. It also i s shown that power numbers for small
closed vessels are greater than for open ones of otherwise identical geometries.
F o r geometrically similar baffled agitated vessels, the power input from the impeller is related to rotational speed and physical properties by the equation
NP = K ( N R ~ ) ‘
(1)
where N p is the power number, P N3D5p, NRe is the Reynolds number, N D L ’ u ,and K and a are dimensionless constants dependent on the system geometry. Above a certain minimum Reynolds number, (NRe)mln, Np becomes constant (Bates et al., 1966). This paper reports values of Np above ( N R , ) ~for ,~ a variety of impeller-vessel geometries as set out in Figure 1. The data were collected as part of a study of solidwater mass transfer but are presented separately because they represent the most comprehensive data available giving N P values for “closed” vessels-i e., with no airwater interface. I n particular, the effect of impeller size and clearance from the base of two sizes of such vessels is investigated for 6-blade disk turbines, 4-blade, 45’ pitch turbines, and 2-blade flat paddles. One set of runs was carried out with an air-water interface to facilitate comparison of the results from this work with those of previous investigations. The experimental equiment has been reported in detail elsewhere (Nienow and Miles, 1969). Basically, the dynamometer consisted of a minimum-friction, combined thrust and journal air-bearing which supported the vessel. The torque was then detected by the angular displacement of the vessel which was opposed by a calibrated helical spring. All the runs were carried out a t 25°C using water as the working fluid and in all (625 data pairs were obtained), each torque reading was the mean of three separate measurements. For each geometry, the maximum impeller speed was that necessary to achieve a Reynolds number of 10’.
general, K p tended to fall very slightly with increasing NRe, probably a result of the effect of enhanced skin friction because of the lid of the vessel. Though few results were obtained for the open top vessel in this work, the power numbers appeared constant unless aeration occurred through the upper surface of the liquid when, as reported previously (Clark and Vermeulen, 1964), the power number fell most markedly. All these “constant” N p values are given in Table I for each system geometry studied. The geometry for each run is identifiable by a letter-number combination. Discussion
As seen from Table I , the variation of power number with system geometry is somewhat erratic. For instance, comparison of the results for the geometrically identical large and small systems shows a very close correspondence between N P values for the 4-blade, 45O-pitch turbine, a
t. +Wade
y
Z-T=b*;N-40
2/.
L -T-l;t
to
2400RW
N = 10 t o
1000 RPM
T
Dirk Turbine
4-Blade 15‘-Pitch
Turbine
2-Blade
Flat Paddle
Results
Power numbers were determined for Reynolds numbers from about 6 x 10’ to lo”, the range of importance for the dissolution studies. I n general, the power number fell a t the beginning of the range, and, though leveling off, never became exactly constant. However, for each impeller above a Reynolds number of 2 x lo4, variations from a constant value were always less than i.105 and usually less than & 5 5 . KO unique variation from constancy was found but, in I
To whom correspondence should be addressed.
D
c
D
1
Figure 1. Closed vessel and impeller dimensions
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
41
Table I. Power Numbers for 2 X lo4 < NRe < 10’ Identifica ti on, run no.
T, in.
Air/water interface
Impeller clearance, C/Z
%(B)
%(A)
?4(C)
%(D)
%(E)
3/4(F)
x/D,
4.1 5.0 5.6 Aerated 5.6 5.9 5.8
3.9 4.8 5.0 Aerated 5.0
3.7 4.7 4.6 Aerated 4.8
0.44 0.33 0.22 0.22 0.14 0.10 0.05
3.0 3.4 3.0 3.3 3.5
2.8 3.2 2.8
2.7 3.0 2.7
1.9 1.6 2.3 1.8 1.7
1.9 1.6 2.1
1.8 1.5 2.0
6-Blade Disk Turbine 1 2 3 4
5 6 7
6 6 6 6 6 12 12
no no no Yes no no no
3.6 4.4 4.6 4.2 4.3
3.8 4.7 4.9 4.8 5.0 5.5 5.5
3.9 4.9 5.3 5.0 5.3
2-Blade Flat Paddles 8 9 10 11 12
6 6 6 12 12
no no no no no
2.7 2.6 2.5
2.8 3.1 2.7 2.8 3.0
2.9 3.3 2.9
4-Blade, 450-Pitch Turbine 13 14 15 16 17
6 6 6 12 12
no no no no no
1.9 1.6 2.3
1.8 1.4 2.2 1.7 1.4
reasonable correspondence for the 2-blade flat paddle, and an approximately 50% difference for the 6-blade disk turbine. Because of this erratic behavior, the data have been presented in tabular form. No satisfactory graphical presentation was apparent. Disk Turbine Results. Consider first the very low N P values obtained in the small vessel with the smallest impeller (set 1) as compared with those for apparently identical geometrical ratios on the larger scale (set 6). In fact, because of the mode of manufacture, strict geometrical similarity could not be maintained except for the major dimensions, as set out in Figure 1. Though the importance of observing very strict geometrical similarity has been pointed out previously (Bates et al., 1963), the magnitude of Np changes with minor dimension changes has not been reported. Though all minor dimension ratios changed very slightly, the major variation was in the carrier disk thickness t o blade width ratio as shown in Table I. As x i D , increases, the friction loss from the inside edge of the blades of the disk turbine would tend to decrease and therefore so would N p . Some measure of the magnitude of the contribution of the inside edge to the total power number can be obtained from the work of Bates et al. (1963). They found that N p for an open, flat-blade turbine was about 25% less than that for a disk turbine of the same overall dimensions, even though the latter has blades only half the length of the former. Assuming that the loss per unit length of blade perimeter from top and bottom edges is the same for each type and also that losses from the inside and outside edges of the disk turbine are equivalent, then the contribution to Np is five times as high from the ends of the blades as from the sides. On this basis, since N P for configuration 6B is 5.5, Spfor configuration 1B should be 4.4 or 3.3, depending on whether it is assumed that the inner edge contributes “half an edge” or “no edge”. Therefore, the major reason for the difference in Np’s between set 1 and set 3 probably is owing to the change in xID,. 42
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
1.7 1.4 1.9
For the impeller used in the set 3 runs ( x / D , = 0.22), the effect of the carrier disk would be much less and in fact in this case the results are in better agreement with the equivalent large system (set 7, x/D,, = 0.05). However, since x / D , is so obviously an important parameter, the Np data for the small system must be used with considerable caution; the absolute values should not be taken as applicable for larger scale work. Only trends resulting from changes of impeller position within the vessel are strictly reliable. Bearing the above discussion in mind, apparently there is reasonable agreement between the currently accepted value of Np for an open vessel ( D / T = 54, C / Z = %) of 5.0 (Bates et al., 1963) and the present result for geometry 4C ( D / T = %, C / Z = f/l) since the increase in Np due to the increase in D / T above % (Bates et al., 1963) would probably be balanced by the reduction in N P owing to a thick carrier disk. This work suggests that there is also an increase in Np with increasing D I T in closed vessels but the effect is masked since a t the same time N P is increased by the reduction of x/D,. The reduction in NP as the base is approached confirms the results of Bates et al. (1963) for open vessels. I n addition, an almost identical reduction was found as the lid was approached (see the results for geometries 2B and 2F and for geometries 5B and 5F). However, for open vessels, Bates et al. (1963) have shown that the proximity of a turbine to the free surface for clearances greater than C / D = 1 would have a negligible effect on power provided aeration did not occur. These results show that in the case of closed vessels, this statement does not hold and that the power number is a maximum when the impeller is placed midway between the top and bottom. Closing the vessel increased the N P value when using the disk-turbine impeller. This can be seen by comparing Np’s for each of the closed geometries of set 3 with the equivalent open geometries of set 4. On the other hand, Laity and Treybal (1957) using 12-in. and 18-in. vessels with geometrical ratios identical to those of run 2D except
for 16.7% baffles, found that Np was constant between about 5.5 and 6.2 for both open and closed vessels. These values are in good agreement with those of this work for the 12-in. closed vessel. Yet in an earlier work, not quantitatively comparable with this because of the low Reynolds number used, Flynn and Treybal (1955) found N p values of 6.0 and 9.5 for open and closed vessels, respectively, again suggesting that closing the vessel increased Np. Though there is some conflict in these results, it seems reasonable that closing the vessel enhances Np and that the data for the 12-in. system is satisfactory for large-scale work. Many workers in agitated systems have used the disk turbine in studies covering a very wide variety of processes; often these studies were on different scales, some of them very small. However, it is very difficult to construct disk turbines with constant x / D , ratios in both very small and large systems. Yet, generally power numbers were not measured and it was assumed that they were constant for all sizes of systems. From these present measurements. obviously, such an assumption could lead to very substantial errors in NP and therefore, in any correlation based on ?Jp. 2-Blade Flat Paddle. For this impeller and also for the 45O-pitch turbine discussed below, very close geometrical similarity was maintained and the results for the equivalent large and small systems were in good agreement. The variation of N P with system geometry is lessi e , 2.5 to 3.5, than with the other types of impeller. Again, for each impeller the maximum value of Np corresponds to a clearance midway between the top and bottom of the vessel. This is in disagreement with the early work in open vessels of Mack and Kroll (1948) who found impeller clearance had no effect. Also, in all cases except with the K clearance, the power number increases as D i T increases from 14 to 3.1. and then falls again as D I T goes from % to 3/1. K O other data for closed systems have been reported. For open vessels with geometrical ratios identical t o 2C (except for the different impeller type) Rushton et al. (1950) reported NP = 1.8 for vessels from 9- to 96-in. diam while Brown (1965) found N P = 2.8 for 6- and 9-in. diam vessels. Thus, in this case, no conclusion can be drawn regarding the effect of closing the vessel. 4-Blade, 450-Pitch Turbine. I n this case, as D I T goes to ‘/r to 3/1, so Np first falls and then increases from again. Bates et al. (1966) reported a fall for propellers in open vessels as D J T increased from 0.33 to 0.4 but the subsequent increase has not been reported.
Bates et al. (1963) found that, for small clearances in open vessels, as the impeller clearance was increased so the power number fell from about 1.9 to about 1.4. This work shows that a similar effect is obtained in closed vessels as C / Z increases from 1% to 5 i . However, in addition, a further increase i? C J Z to ?< produces a sudden increase in NP while a still further increase again causes N P to fall. Again no other results are available for closed systems. For open systems with geometrical ratios identical to those of 2C except for impeller type, Rushton et al. (1950) found N P = 1.2. Apparently the closing of the vessel caused an enhancement of N P but no definite conclusion should be drawn. Nomenclature
a = constant, dimensionless baffle width, L impeller clearance above the tank bottom, L impeller diameter, L length of turbine blade, L width of impeller blade, L , see Figure 1 constant, dimensionless impeller speed, T-I power number, P / p N 3 D b ,dimensionless Reynolds number, ND’iv dimensionless impeller power, M L T ’ tank diameter, L u! = 45O-pitch turbine blade width, L , see Figure 1 x = carrier disk thickness, L , see Figure 1 Z = liquid height, L p = fluid density, ML-’ Y = fluid kinematic viscosity, L’T-’
B = C = D = Dr. = D,, = K = N = NP = NRe = P = T =
Literature Cited
Bates, R. L., Fondy, P. L., Corpstein, R. R., IND. ENG. CHEM.PROCESS DES. DEVELOP. 2, 310 (1963). Bates, R. L., Fondy, P. L., Fenic, J. G., in “Mixing,” Uhl, V. W., Gray, J. B., Eds.. Vol. 1, Ch. 3, p 129, Academic Press, New York, 1966. Brown, D. E., Ph.D. Thesis, University of Newcastle, 1965. Clark, A. W., Vermeulen, T., AIChE J . 10, 420 (1964). Flynn, A. W., Treybal, R . E., A I C h E J . 1, 324 (1955). Laity, D. S., Treybal, R . W., AIChE J . 3, 176 (1957). Mack, D. E., Droll, A. E., Chem. Eng. Progr. 44, 189 (1948). Yienow, A. W., Miles, D., J . Sci. Instrum. 2 (a), 994 (1969). Rushton, J. H., Costich, E. W., Everett, H. J., Chern. Eng. Progr. 46, 395, 4467 (1950). RECEIVED for review December 1, 1969 ACCEPTED September 3, 1970
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