858
Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 858-867
scription, eq 1 B = constant system matrix in the linear state space description, eq l C = constant system matrix in the linear state space description, eq l D = constant system matrix in the linear state space description, eq l D = distillate flow rate d = disturbance vector in eq 1 F = robustness filter in eq 15 f , = feed flow rate to stage j , kg-mol/h = transfer-function matrix of the plant G = transfer-function matrix of the model G , = controller transfer function matrix in the Internal Model Control structure I= unit matrix L = multiplicative error matrix L j = liquid flow rate leaving stage j , kg-mol/h Lo = multiplicative error matrix acting on outputs 1 = uncertainty bound of the plant m = last stage of the rectifying column n = number of distinct singular values N = total number of stages Q, = condenser duty, W Q, = reboiler duty, W Th= reboiler heating medium temperature, K U = unitary matrix composed of the left singular vectors UA = heat-transfer parameter, J/(s K) u = input vector in eq 1 ll4lmax 7 upper bound of ll4l V = unitary matrix composed of the right singular vectors VI = vapor flow rate leaving stage j , kg-mol/h x = state vector in eq 1 x, = liquid composition at stage j y = output vector in eq 1 ys = set-point vector z j = feed composition to stage j
4
Greek Letters y = sensitivity function defined by X, = eigenvalues , ,X = maximum eigenvalue Amin
eq 16
= minimum eigenvalue
ui = singular values
a = maximum singular value minimum singular value w = angular frequency, rad/h
u =
Superscripts H = complex conjugate transpose L i t e r a t u r e Cited Doukas, N. P.; Luyben, W. L. Ind. Eng. Chem. ProcessDes. Dev. 1881,20 (I), 147. Doyle, J. C.; Stein, G. IEEE Trans. Autom. Control 1881, AC-26 (I), 4. Fitzmorris, R. E.; Mah, R. S. H. A I C M J . 1880, 26(2),265. Garcia, C. E.; Morari, M. Ind. Eng. Chem. Process D e s . D e v . 1982, 21 (2), 308. @Osdidier, P.; Morari, M.; Holt, B. R. Ind. Eng. Chem. Fundam., in press. Holt, B. R.; Morari, M. Chem. Eng. Scl., in press. HoA, 8. R.; Morari, M. Chem. Eng. Sci., in press. Klema, V. C.; Laub, A. J. IEEE Trans. Autom. Control 1880, AC-25 (2), 164. MacFarlane, A. G. J.; Scott-Jones, D. F. A. Int. J . Control 1978. 29 (l), 65. Mah, R. S. H.; Nicholas, J. J., Jr.; Wodnik. R. B. AIChE J. 1877, 23 (9,651. Morari. M. IEEE Trans. Autom. Control. in press. Morari, M.; Gri", W.; Oglesby, M. J.; Prosser, I. D. Chem. €ng. Sci., in press. Morari, M. Chem. Eng. Sci. 1883, 38, 1981. Rosenbrock, H. H. "Computer-Aided Control System Design"; Academic Press: London, 1974. Shimizu, K.; Mah, R. S. H. " p u t . Chem. Eng. 1983a, 7(2), 105. Shimiru, K.; Mah, R. S. H. Comput. Chem. Eng. 1883b, 7(2), 123. Shimizu, K.; Mah, R. S. H. Paper presented at the Proceedings of the American Control Conference, Sen Francisco, CA, 1 9 8 3 ~p 200. Soave, G. Chem. Eng. Sci. 1972, 27 (6), 1197. Tyreus, B. D.; Luyben, W. L. Chem. Eng. f r o g . 1876, 72(9). 59.
Received for review June 7 , 1984 Revised manuscript received November 12, 1984 Accepted November 21, 1984
Impeller Power Numbers and Impeller Flow Numbers in Profiled Bottom Tanks Michael W. Chudacek M. D. Research Company Pfy. Ltd., North Ryde 21 13, Australia
Impeller power numbers and impeller flow numbers are reported as a function of impeller off-bottom clearance for three types of bottom geometry: fully profiled, cone and fillet, and the classical flat bottom design. The experiments were primarily conducted in 0.5-m scale models of the above geometries, but some data were also collected in I-m models. The 3-blade-square-pitch propeller was the principal Impeller examined, but some data were collected by using 6-inclined-blade turbine and dual impeller systems. Results for the flat bottom tank are compared with those reported by other authors. The differences betweenthe data reported in various mixing studies are attributed to possible errors in torque measurement and more importantly to minor differences in vessel and impeller geometry. I t is proposed that for an unequivocal description of the propeller, projected blade area and blade height to width ratio should be reported In addition to the propeller diameter and pitch.
impeller in a given geometry.
Introduction
The impeller power number, Np,represents an important parameter used for calculation of impeller power requirements for mixing of liquids or suspensions in the turbulent region. The impeller discharge flow number, Nqd,indicates the pumping capacity of the impeller in a given geometry. The impeller power number to impeller discharge flow number ratio gives information about pumping efficiency of the
Power Number Theory
The use of dimensional analysis for correlating impeller power was initially suggested by White and co-workers (1934a,b; 1936). This approach was further developed by Hixson and Luedeke (1937)and Rushton et al. (1950). Bates et al. (1963)reported a power number relationship using impeller diameter, d , as the reference length
0196-43a5/wii24-aa5a$a~ .5wo 0 1985 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 859
5
where the first term is known as the tank Reynolds number Re and the second term is known as the Froude number. The Reynolds number describes the hydrodynamic effects in the system. The Froude number accounts for vortexing effects in the system. The remaining terms account for the effects of the vessel geometry and impeller geometry. Bates et al. (1963) also pointed out that the equation should be expanded to include baffle number and width, spacings of multiple impellers, and off-center impeller positioning. All of these additional parameters may be included in a form similar to the geometrical terms of eq 1or in the form of a factor as used by Bates et al. (1963). At this point it should be noted that the effect of the tank bottom shape has never been included in the above analysis. The tank bottom shape represents a significant geometrical factor with respect to the recirculation pattern and is also likely to influence the impeller power number. At present, besides the classical flat bottom, there are numerous tank bottom shapes used by industry and research such as torispherical (dished), hemispherical, and conical types. Also, profiled bottom tanks of the types outlined in Figures 1and 2 and shown by Chudacek (1983) to be highly efficient for solids suspension may gain industrial acceptance. Chudacek (1984) reported optimized dimensions for these profiled bottom geometries, matched to an impeller of given diameter, and there is no practical reason for using geometries outside these optimal dimensions. AU presently used and proposed tank bottom geometries exhibit vast differences in design so it is unlikely that the bottom shape can be grouped as a single geometric parameter. In these circumstances a comprehensive relationship describing parameters of bottom shape would not only be fairly complex but also of limited value. With new designs of profiled bottom tanks gaining acceptance, the design engineer requires some power number correlations for these systems. A convenient way to obtain these power number correlations is to incorporate the effects of bottom shape into the coefficient k of eq 1. There are also likely to be some differences in exponents of the various geometric or even the hydrodynamic terms in eq 1 for significantly different geometries. While eq 1has a fundamental significance, it is seldom required in its full form. For the case of a fully baffled (b = 0.m)turbulent mixing regime of homogeneous liquid, Rushton et al. (1950) reported an almost constant power number for a 3-blade propeller for Re 1 lo4,and constant power number for an 8-inclined-blade turbine (459for Re L 2000. For a given geometry in the above regimes, eq 1 reduces to D
Therefore eq 2 may be used to calculate impeller power input for homogeneous liquids if the power number of the impellers is known. Impeller Discharge Flow Number The impeller discharge number Nqd defined by eq 3 indicates the pumping capacity of the impeller. The (3)
T
1
T = d = b = a = f = Rp = hp = rp = b, = db = ab = C
D 0.33D 0.10D 0.02D 0.02D 0.25D 0.15 D 0,030 0.15 D 0.30D 0.01D = 0.166D to 0.50D
Figure 1. Geometrical parameters of fully profiled bottom tank
(PBT). discharge number has been reported constant in the turbulent region below the aeration threshold by Porcelli and Marr (1962), Nagata (1975), and Medek and Fort (1975). Impellers are sometimes rated by their power number to discharge flow number ratio Np/Nqd.This ratio is an indicator of pumping efficiency of an impeller-the smaller the ratio, the higher the pumping efficiency. A wide range of N p values appears in the literature for various impellers in the Rushton tank geometry outlined in Figure 3. The influences of impeller clearance, impeller diameter, and tank bottom geometry are seldom reported. Impeller power numbers and discharge numbers are often reported without reference to the tank geometry used. These omissions lead to unnecessary confusion and inaccuracies in the prediction of impeller power requirement and pumping capacity. This work has the following aims: (1)to show that the power number and discharge number are dependent on impeller clearance, impeller diameter, tank height, tank bottom shape, and impeller blade geometry; (2) to describe the relationships between tank geometry and power and discharge numbers and explain observed trends; (3) to provide tables of precise values of power numbers and discharge numbers for the flat bottom tank and alternative geometries; (4)to compare power number values found in this work with those obtained by other authors and discuss the discrepancies.
Experimental Equipment and Procedure Experiments were conducted principally in 0.5-m diameter models of the three tank geometries under investigation. Some experiments were also conducted in 1.0-m models. The models examined were the fully profiled bottom tank (PBT) defined by Figure 1; the cone and fillet bottom tank (CFT) defined by Figure 2; and the standard flat'bottom tank (FBT) defined by Figure 3. All tanks had four baffles and were agitated by a 3-blade-squarepitch propeller (3BSPP) defined by Figure 4. Data were also collected for a 6-inclined-blade turbine (GIBT) (45') with dimensions as given in Figure 5.
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
T = D d = 0.33D b = 0.10D a = 0.02D f = 0.02D d , = 0.30D h i = 0'20D d2= 0'7QD h2= 0.15D C = 0,2510 0.50D e = 0.250
T
Figure 2. Geometrical parameters of cone and fillet tank (CFT).
I-
ui
t
Figure 5. Geometrical parameters of &inclined-bladeturbine (45O) (6IBT).
Figure 6. Isometric view of the profiled tracer, similar to Steidl (1958).
L
Figure 3. Geometrical parameters of standard flat bottom tank
(FBT). 10
20
30
, 40"
,,,\
4
0 10 20 30 40"
4'
f
T = D d = 0.33D b = 0.10D a = 0.02D f = 0,020 c = 0,083~ to 0.33D
Ti
0
*
0.012 d
7 - 7 d=O 3BSPP 333
Pitch-1 0 3 -~ -
~
~
I
1
_.__~
Figure 4. Blade cross section of 3-blade-square-pitch propellers used in 0.5-m and 1.0-m tank models. 3BSPP d = 0.1667 m, x A = 0.462, h / w = 0.512, C, = 0.483. 3BSPP: d = 0.333 m, xA = 0.452, h / w = 0.539, C, = 0.532.
The tanks were constructed entirely from transparent materials to allow observation of a recirculation tracer. Impeller speed was adjustable in the range 80-1450 rpm and was monitored by a magnetic sensor on the drive shaft with f0.5-rpm resolution. The experiments were conducted with filtered water, thermostatically controlled at 25.0 i0.2 "C. Only a highly
turbulent range below the aeration threshold Re 1.0-2.5 X lo6 was examined. This Reynolds number range was of significant interest to us as it relates to solids suspension in the studied geometries. Various impeller off-bottom clearances were tested in the ranges indicated in Figures 1,2, and 3. The minimum clearance permitted by the bottom geometry was 0.1660 in the fully profiled bottom tank and 0.2500 in the cone and fillet tank. Tank heights of 1.00 and 0.80 were examined with the cone and fillet geometry. Also, the fully profiled bottom tank was examined by using 3-bladesquare-pitch propellers of 0.405 and 0.5100 diameter at various clearances (0.166-0.50). Dual 3-blade-square-pitch propellers located one above the other were also examined. In this case the lower propeller was located at off-bottom clearance c = 0.250 and the upper propeller was located at depth e = 0.250. Power numbers were obtained from torque measurements with a precalibrated strain gauge bridge mounted on a torsion element in the drive shaft. The strain gauge assembly was calibrated before and after the experiments on a separate calibration rig. No-load dynamic friction in the impeller shaft was determined over the whole experimental range and monitored at regular intervals. Torque measurements were corrected for the no-load dynamic friction in the impeller shaft, and the resulting net torque error at all speeds did not exceed *2%. All power number experiments were conducted in duplicate, and at least 11 data pairs were collected per experiment. The reproducibility was better than &l.O%. The discharge numbers were determined by the tracer counting method reported by Porcelli and Marr (1962), using a profiled tracer as reported by Steidl (1958) and shown in Figure 6. Throughout the experimentation, three 15-min counts were made at each of three impeller speeds, and the reported values of discharge numbers are the mean values obtained at these three speeds.
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 861
Table I. Impeller Power Numbers for the 6-Inclined-Blade Turbine 45O in Various Geometries impeller clearance, D eeometrv tank size. m imD. no. 0.083 0.166 0.250 0.333 FBT 1.0 1 2.07 1.92 1.75 1.63 FBT 0.5 1 2.11 1.90 1.73 1.63 FBT 0.5 2" 2.02 1.84 1.65 1.53 PBT 0.5 1 1.87 1.73 1.63
0.416
0.500
1.59
1.55
0.416
0.500
0.316 0.353 0.358
0.334 0.374 0.377
"Turbine blades set accidentally to 43-44', Table 11. Impeller Power Numbers for the 3-Blade-Square-Pitch Propellers in Various Geometries impeller clearance, D geometry tank size, m imp. no. 0.083 0.166 0.250 0.333 FTB 1.0 1 0.410 0.355 0.314 0.325 FTB 0.5 1 0.320 0.361 0.366 0.340 CFT 1.0 1 0.331 0.317 0.359 0.342 CFT 0.5 1 PBT 0.5 1 0.385 0.355 0.347
For the single impeller the discharge number Nq! was calculated from the number of downward passes within the impeller perimeter by using the relationship
VN2 Nqd = (4) td3N where V is the volume of the tank, N2 is the number of passes, N is the impeller speed, t is the counting period, and d is the impeller diameter. The downward passes outside the impeller perimeter N4 were used to calculate the induced flow number N .. The sum of the discharge number Nqd and the induce3 flow number Nqi is defined as the recirculation flow number Nqr. For the dual impeller arrangement, four different tracer passes were recorded: first, passes within the perimeter of the upper impeller N l ; second, passes within the perimeter of the lower impeller N,; third, passes from the upper impeller into the bulk of the tank due to a collision of the tracer with the impeller N3;fourth, downward passes outside the lower impeller perimeter N4. These passes yielded a discharge number from the upper impeller Nqdl, a discharge number from the lower impeller Nqd2,and an induced flow number Nqi as shown below. Nqdl Nqd2
Nqi
a N1
N2 a
+ N3
N4
(5)
(6) (7)
Results and Discussion Impeller Power Numbers. The form of eq 1suggests a log/log plot as the appropriate method of presentation for power number data in relation to other parameters. This method of presentation has been used by many workers and is correct with respect to dimensional analysis. It will also smooth out scattered data. However, the objectives of the present study call for precise power number data to be provided for the geometries studied and for even weak trends to be identified. The data are therefore presented in a tabular form and in expanded Cartesian plots rather than log/log plots. 6-Inclined Blade Turbine. The impeller power numbers for the 6-inclined-blade turbine 45O (GIBT, defined by Figure 5) at different off-bottom clearances are given in Table I. The first two lines in Table I show excellent agreement of the power numbers obtained for the 1.0- and 0.5-m tanks. This suggests that the two tank sizes can be considered hydrodynamically similar. Comparing lines two and four, the relationships between power number and impeller off-bottom clearance for the
1
1
Geometry
,
,;:
CFT CFT
Liquid height [0]
0.8
A
0.2 0.1
0.2 0.3 0.4 Impeller clearance [o]
0.5
Figure 7. Power numbers for the 3-blade-square-pitchpropeller vs. impeller off-bottom clearance for various geometries.
6IBT are slightly different for the FBT and the PBT. The power number plotted against impeller clearance in log/log coordinates yielded an exponent x 5 = 0.20 (eq 1)for the FBT in the 0.083-0.3330 clearance range and x 5 = 0.18 for the PBT in the 0.250-0.50 range. Medek (1978) reported a value x 5 = 0.16 for the GIBT in the FBT, in reasonable agreement with the present data. Discharge flow from the impeller is hindered (i-e., pumping head increases) when any solid surface is located in the discharge flowpath close to the impeller. This hindrance in turn causes the impeller power number to increase. The relatively minor difference in power numbers for the FBT and PBT suggests that the central profile has little influence on the discharge flow from the 6IBT. The GIBT has a much wider discharge flow than the 3BSPP due to a radial flow component, and flow from this impeller practically misses the central profile. Given these flow characterisitics, the central profiles of the PBT and CFT would be expected to affect recirculation efficiency much less in tanks agitated by the GIBT than in those agitated by the 3BSPP. The wider discharge flow from the GIBT would also suggest that the central region of the tank bottom has less intense agitation than its periphery. The GIBT is therefore not recommended for solids suspension agitation in the PBT as a solids fillet is likely to form at the base of the central profile. 3-Blade-Square-PitchPropeller. Power numbers for the 3BSPP with various impeller off-bottom clearances and tank geometries are given in Table I1 and plotted against impeller clearance in Figure 7. From Figure 7 it can be seen that the power numbers are higher for the CFT and PBT than for the FBT at
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Table 111. ImDeller Power Numbers for t h e 6-Inclined-Blade Turbine in the Flat Bottom T a n k Reported by Various Authors impeller clearance, D authors biD wld Dl m 0.166 0.250 0.333 0.500 Medek (1978) 0.1 0.2 0.3 1.74 1.64 1.56 1.40 0.4 0.6 Seichter (1975) 0.1 0.2 1.58 0.1 0.2 0.43 1.65 Rushton et al. (1950) 0.2 0.5 1.91 1.74 1.63 present work 0.1 1.0
d = 0.405 D d=0.510D
0.4
I
0 0
0.
0.1
0.3 0.4 0.5 Impeller clearance [D]
0.2
Figure 8. Power numbers for 3-blade-square-pitch propeller in the PBT as function of impeller clearance and impeller diameter.
clearances below 0.330. Also, power numbers are higher for the CFT than for the PBT at 0.250 impeller clearance. These power number differences can be attributed to the different flow deflecting characteristics of the respective profiles. The deflection by t;he profiles causes some restriction to the flow (increased head) and thus a rise in power number. From Figure 7 it can also be Seen that lowering the liquid level in the CFT tank from 1.00 to 0.80 markedly reduced the power number, on average by 9.4%. In contrast, Medek (1978) reported only a very small decrease in power number for a similar reduction in liquid level in the FBT agitated by the 6-inclined-blade turbine (45O). This suggests that recirculation patterns in both geometries are not the same. Figure 8 presents the power number for the 3BSPP in the PBT as a function of clearance c and propeller diameter d. This clearly shows that the power number decreases significantly as the impeller diameter increases. Similar results were reported by Medek (1978) and Seichter (1975) for the GIBT (45O) in the FBT. Figure 8 also indicates that the power number vs. impeller clearance relationship becomes steeper as the impeller diameter increases.
Comparison with Other Work Power numbers reported for the GIBT (45O) by other authors are given in Table 111. The power number for the GIBT at 0.3330 clearance reported by Rushton et al. (1950) was 1.2% higher than our value, while the values reported by Medek (1978) for several clearances were smaller by 4.3-8.9%. The discrepancy increased with decreasing clearance. The difference in compared power numbers is considerable, and an analysis of the possible sources of error is worthwhile. The errors encountered in power number estimation can be loosely grouped into two categories.: (i) small errors in torque measurement; (ii) slight geometrical differences in the system studied.
‘Withrespect to torque measurement, two types of errors may be encountered. First, in systems using a dynamometer or a strain gauge mounted on the drive shaft for torque measurement, the torque is likely to be overestimated unless the static and dynamic friction in the shaft bearings is accounted for. Second, in systems where torque is transmitted via an arm from a suspended “freely” rotating system to a sensor, the measured torque will be underestimated if the static and dynamic friction losses of the suspension assembly are not accounted for. With respect to geometrical parameters, the apparatus used by Medek (1978) and Seichter (1975) had 0.10 wide baffles positioned in contact with the tank wall. Our geometry used baffles of the same width, but positioned at a distance 0.02D from the tank wall for the purpose of facilitating solids suspension. These baffles may thus have been effectively 0.120 wide, and this may in turn have caused a slight rise in the power number. Rushton et al. (1950),however, used a geometry identical to Medek (1978) and Seichter (1975),and his results were very close to those of the present work. Strict geometrical similarity of impellers is an essential prerequisite for accurate scale-up study. For example, the thickness of the blades is not frequently reported, despite its influence on power number as shown by Medek (1978). In addition, our experience has shown even minute variations in blade angle that cannot be distinguished visually can be a source of error. Lines 2 and 3 in Table I show power numbers for a GIBT with blades set at exactly 45’ and a turbine of identical diameter with its blades set at 43-44O due to a manufacturing error. The erroneous turbine gave a 4.6% lower value of power numbers that could not be otherwise explained. This highlights the importance of precision manufacturing of impellers used for fundamental studies of mixing. Similar small geometrical inconsistencies in vessels of different scales may be a contributing factor to the lack of agreement of data for scale-up relationships reported by various authors. Geometrical discrepancies much more significant than those described above were encountered with the 3BSPP’s used in the 0.5- and 1-m model tanks. In Table 11, agreement of power numbers for these impellers is clearly not as good as those reported for the GIBT. The power numbers in the 0.5-m CFT were found to be 10% higher than in the 1-m CFT. A similar discrepancy was also found for the FBT. The pitch of both the test propellers was close to unity, and the blade width was also to scale. The only other factors possibly influencing the power number could be the blade camber, the blade height, and the projected blade area. These parameters are seldom reported in mixing studies. However, the following example illustrates their importance. In a set of five investmentcast stainless steel industrial propellers (d = 0.1667 m) from a well-known manufacturer no two propellers had equivalent power numbers, and the power number varied by *8.2%. Careful measurement revealed only slight variations in pitch (f3%) between different propellers and negligible variations in blade width (fl%). The variations
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 863 Table IV. Impeller Discharge Flow Numbers for 6-Inclined-Blade Turbines and 3-Blade-Square-Pitch Propellers in Various Geometries impeller clearance, D impeller geometry side baffles bottom baffles 0.166 0.250 0.333 0.416 0.500 0.98 6IBT(45O)' FBT 4 0.78 0.99 0.90 0.61 61BT(45')* FBT 4 6IBT(45') PBT 4 1.19 1.30 1.27 1.01 0.89 3BSPPc FBT 4 0.60 3BSPPd FBT 4 0.61 3BSPP FBT 4 0.58 0.60 0.61 0.60 0.55 3BSPP PBT 4 0.57 0.63 0.67 0.64 0.62 3BSPP PBT 4 4 0.60 0.62 0.67 0.65 0.61 3BSPP PBT 4 0.56 0.58 0.39' 3BSPP CFT 4 0.59 0.61 "Fort & Sedlakova (1968). *Medek & Fort (1975). cVan de Vusse (1955). dMarr & Johnson (1963). evortex formation.
in power number correlated poorly with the variations in pitch and blade width. Measurements of blade camber, however, revealed great variations gnd irregularities, even from blade to blade on the same propeller. To clarify the above-described discrepancy the power numbers of the five 3BSPP tested with d = 0.1667 m and one "matching" 3BSPP with d = 0.333 m were correlated against propeller pitch, fraction of projected blade area xA given by eq 8,
blade height to blade width ratio, and camber coefficient C, given by eq 9,
c, = A, - 1000 d2 where Ab is projected blade area, dh is impeller hub diameter, and A, is area under the concave curve in the widest cross section of the propeller blade (see Figure 4). For the CFT with liquid height T = 0.8D agitated by the 3BSPP at 0.3330 impeller off-bottom clearance, the best correlation (correlation coefficient 0.983) was
This correlation has been estimated for projected area fraction range 0.452-0.493, blade height to blade width ratio range 0.512-0.539, and camber coefficient range 0.001-1.515. A similar correlation using propeller pitch in the range 0.985-1.059 instead of blade height to blade width ratio yielded a correlation coefficient of only 0.939. This latter correlation is therefore not reported as it is clearly inferior to that given by eq 10. There are two reasons likely for the poor correlation coefficient using propeller pitch instead of blade height to width ratio. First, it is extremely difficult to obtain propeller pitch measurement on the tip of the propeller blade. Second, industrial propellers are manufactured, as has been shown above, to low tolerances so constant pitch can hardly be expected. In contrast, the propeller blade height and blade width are easily measured. These parameters are measured on the widest c r m section of the blade, which is likely to have the greatest effect on propeller power number. The use of propeller blade beight to blade width ratio (a measure of the pitch a t the widest part of the blade) is therefore more apropriate for description of the propeller than the use of propeller pitch measured at the tip of the blade.
The differences in correlation coefficient shown above clearly support this view. The above result indicates again how much error can be caused by not using strictly geometrically identical propellers for scale-up studies. So far, it has been customary in mixing studies to report the propeller diameter, number of blades, and pitch. Only a few investigators have reported projected blade area. It seems that reporting of the cross section at various radii of the blade is essential for full definition of the propeller. From this point of view standardization of industrial propellers would be helpful. At present a comparison of propeller power numbers is of only limited value, because the exact propeller geometry for mixing appplication remains to be satisfactorily defined. The study of marine propeller design literature may provide an answer. The problems associated with acquisition of defined and geometrically similar propellers may be the reason why such very effective impellers were relatively infrequently used in fundamental studies of mixing. The only reference found was that of Rushton et al. (1950))who reported for a 3-blade-square-pitch marine propeller in the FBT with four baffles 0.1D wide and impeller clearance of 8.3330, a power number 0.32. This value is in good agreement with values of 0.340 for the 0.5-m FBT and 0.325 for the 1.0-m FBT obtained at identical clearance in the present study, especially when the ambiguity of the present propeller definition is considered. Discharge Flow Numbers. Details of the tracer method for estimating impeller discharge numbers were given by Porcelli and Marr (1962) and will not be repeated here. During preliminary experiments two types of neutrally buoyant tracers were tested-a plain spherical tracer (diameter 8 mm) and a profiled spherical tracer (diameter 8 mm) as reported by Steidl (1958). The profiled sphere gave a 4.8% higher recirculation rate than the plain sphere. The profiled sphere seemed to be better entrained by the flow and was therefore used for the rest of the study. Impeller discharge numbers for the FBT, PBT, and CFT found in this work are given in Table IV, together with FBT data reported by previous authors. Selected cases are plotted in Figure 9. 6-Inclined Blade Turbine. Figure 9 shows the relationships between the 6IBT discharge number and impeller clearance for the PBT in the present work and for the FBT as reported by Medek and Fort (1975). Both cases follow a similar trend, with maximum discharge number occurring at 0.2500 clearance; however, the discharge number in the PBT was 30-50% higher than Medek and Fort's values for the FBT. This quite large difference cannot be solely explained by the smoother recirculation pattern of the PBT compared with the FBT. It seems that Medek and Fort (1975) used a slightly dif-
884
Id.Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
Table V. Impeller Induced Flow Numbers for 6-Inclined-Blade Turbine and 3-Blade-Square-Pitch Propellers in Various Geometries impeller clearance, D impeller geometry side baffles bottom baffles 0.166 0.250 0.333 0.416 0.500 0.65 0.64 0.58 0.40 GIBT PBT 4 0.63 0.49 0.50 0.45 0.39 3BSPP FBT 4 0.48 0.42 0.46 0.46 0.43 3BSPP PBT 4 0.40 0.33 0.07" 4 0.28 3BSPP PBT 0.40 0.45 0.46 0.46 3BSPP PBT 4 4 0.34 0.49 0.55 3BSPP CFT 4
-
vortex formation. Impeller BIBT(45') 61BT(45') 3BSPP BBSPP 3BSPP 3BSPP
zz 0 . 9 1 k o.*L
Geometry FBT PBT FBT PBT PBT PBT
- +-"
..,,
' ,+
+ .,
Baffles Side only x Side only + Side only Side only Side & bottom 0 Bottom only 6
?,..' \
Figure 10. Schematic flow pattern in a baffled flat bottom tank agitated by propeller (Baffles not shown). 0.1
0.3 0.4 0.5 Impeller clearance [D]
0.2
Figure 9. Discharge flow numbers for the FBT and PBT agitated by the GIBT (45') and 3BSPP vs. impeller clearance. Data for GIBT in FBT from Medek and Fort (1975).
ferent method to estimate impeller discharge flow numbers. The GIBT has a wide discharge jet that is likely to be very efficiently diverted by the peripheral profile of the PBT. A t lower clearances the discharge flow is directed to the proximity of the tank bottom to wall junction of the FBT where it is diverted partially up the tank wall and also back along the tank bottom towards the center. This flow diversion in the FBT is much less efficient than the diversion by a smooth profile of the PBT, and this accounts for higher discharge numbers in the PBT. 3-Blade-Square-Pitch Propeller. The 3BSPP discharge numbers were also found higher in the PBT than in the FBT but by a margin of only 5-12.770. The higher discharge numbers in the PBT can be explained by streamlining the PBT tank bottom. The impact of bottom baffles on the discharge number in the PBT was also tested. The use of four bottom baffles in addition to four side baffles (see Figure 1)increased the discharge number by 5% at the lowest clearance only. A t other clearances the discharge number remained unchanged. Four bottom baffles without side baffles yielded, within experimental error, the same discharge number at the lowest impeller clearance as four side baffles. The PBT with bottom baffles only, at 0.2500 impeller clearance, yielded an 8% lower discharge number than the side baffles only configuration. At 0.3330 clearance there was a marked drop in discharge number accompanying vortex formation. From the above it can be said that bottom baffles are efficient in the PBT only at the lowest impeller off-bottom clearance. Simultaneous use of bottom and side baffles increases the discharge number in the PBT only margin-
ally; therefore the addition of bottom baffles is not warranted. Impeller Induced Flow Numbers Impeller induced flow numbers for various geometries and impellers are given in Table V. With one exception, all geometries agitated by the 3BSPP exhibited a weak maximum in the induced flow number at 0.3330 impeller clearance. The exception was the PBT with bottom baffles only, in which the maximum induced flow number was at 0.2500 clearance. It is interesting to note that due to vortex formation there was dramatic drop in the induced flow number in the PBT geometry with the impeller at 0.3330 clearance. For the GIBT in the PBT the maximum induced flow number was also at 0.2500 clearance. The above-described maximums coincide with the maximums for the discharge flow numbers (see Figure 9) as expected. In the PBT, the GIBT and 3BSPP exhibited different induced flow number to discharge flow number ratios. The mean value of the ratio at all clearances was 0.51 f 0.04 for the GIBT and 0.69 f 0.02 for the 3BSPP. This observation suggests that the discharge flow from the BBSPP more efficiently entrains liquid into circulation than that from the GIBT. The improved flow entrainment can be explained by the fact that the GIBT with its broad discharge jet (radial component) upsets the toroidal circulation pattern (see Figure 10) induced outside the truly axially discharging impeller. Any disturbance of this pattern would result in a lower induced flow number. It is also interesting to note that the induced flow number to discharge flow number ratio for the CFT was 0.89 f 0.06, which is some 29% higher than for the PBT. This seemingly paradoxical behavior can be explained as follows. Chudacek (1983) has shown that the central cone in the CFT diverts the flow more efficiently than the central profile in the PBT. This better diversion efficiency provided by the CFT causes the upward flow to be greater
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 865 Table VI. Impeller Recirculation Flow Numbers for 6-Inclined-Blade Turbine and 3-Blade-Square-PitchPropellers in Various Geometries impeller clearance, D impeller geometry side baffles bottom baffles 0.166 0.250 0.333 0.416 0.500 GIBT PBT 4 1.82 1.95 1.91 1.60 1.29 3BSPP FBT 4 1.06 1.09 1.11 1.05 0.94 3BSPP PBT 4 0.97 1.05 1.12 1.10 1.06 3BSPP PBT 4 0.84 0.91 0.46O 3BSPP PBT 4 4 0.93 1.02 1.13 1.12 1.07 3BSPP CFT 4 1.09 1.16 a
Vortex formation.
Table VII. Impeller Power Number and Discharge Number Ratios for Various Impellers and Geometries impeller clearance, D impeller geometry side baffles vessel size 0.166 0.250 0.333 0.416 6IBP FBT 4 0.25 2.23 1.66 1.73 0.45 GIBT PBT 4 0.5 1.57 1.33 1.28 1.58 3BSPP FBT 4 0.5 0.62 0.56 0.56 0.5 0.68 0.56 0.52 0.56 3BSPP PBT 4 0.61 0.56 3BSPP CFT 4 0.5
0.500 2.30 1.74 0.61
aData from Medek & Fort (1975) and Medek (1978). Table VIII. Impeller Power Number to Impeller Recirculation Flow Number Ratios for Various Geometries imDeller clearance. D impe11er geometry side baffles 0.166 0.250 0.333 0.416 GIBT PBT 4 1.03 0.89 0.85 0.99 0.33 0.31 3BSPP FBT 4 0.34 0.34 0.31 0.33 3BSPP PBT 4 0.40 3BSPP CFT 4 0.33 0.31
in this geometry than in the PBT. Therefore the flow induced by this upward flow must also be greater in the CFT than in the PBT. Impeller Recirculation Numbers Impeller recirculation numbers for various geometries and for GIBT and 3BSPP impellers are given in Table VI. The maximum recirculation number for geometries agitated by the 3BSPP was found when the propeller was at 0.3330 clearance. An exception was again the PBT with bottom baffles only, where the maximum was at 0.2500 clearance. It is interesting to note that the CFT gave the highest recirculation number of all geometries agitated by the 3BSPP. For the GIBT in the PBT, the highest recirculation number was found at 0.2500 impeller clearance. These findings support the accepted premise that for the highest recirculation the 3BSPP should be located at 0.3330 clearance. However, results for the GIBT suggest that this impeller yields the highest recirculation when located at 0.2500. Power Number to Flow Number Ratios Power number to discharge flow number ratios for the FBT, PBT, and CFT geometries are given in Table VII. The ratios obtained for the 3BSPP are clearly much smaller than for the 61BT. For example, the ratio for the 3BSPP in the PBT at 0.3330 cleqance is approximately 2.5 times smaller than that for the GIBT. This suggests that the 3BSPP has much higher pumping efficiency than the 6IBT. The minimum value of the ratio for both impellers in this work was found to be at 0.3330 impeller clearance. Medek and Fort (1975,19781, however, found that for the FBT the minimum ratio was at 0.2500 impeller clearance. Power number to recirculation flow number ratios are shown in Table VIII. From this table it can be seen that the GIBT is much less efficient in generating recirculation flow than the 3BSPP. It is most interesting to note that
0.500 1.20 0.36
the power number to recirculation flow number ratios are practically the same at the intermediate clearances of 0.250 and 0.3330 for all geometries agitated by the 3BSPP. Only the PBT at the minimum clearance shows about 18% higher ratio than the FBT. There are two reasons for this higher ratio. First, the power number for the PBT at this clearance is about 7% higher than that for the FBT. Second, the recirculation flow number for the PBT is about 8% lower than for the FBT. These differences in power number and recirculation flow number can be attributed to hindrance of the flow by the proximity of the central profile of the PBT to the lower edge of the impeller. The ratios show a weak minimum at 0.3330 impeller off-bottom clearance. Impellers located at this clearance would therefore provide the most efficient recirculation flow. It is essential to realize that the equal ratios for all geometries do not mean that these geometries would perform equally well for any mixing task governed by recirculation flow. While the compared geometries may show equal ratios, the character of the flow in these geometries is likely to be different. For example, Chudacek (1983) has conclusively shown that the CFT can be up to 140% more efficient for the attainment of complete offbottom suspension of solids than the FBT. Also the FBT is known to have “dead zones” that are eliminated in the CFT. Therefore it would not be surprising if the CFT geometry was found to be superior to the FBT even for liquid blending applications. If taken in isolation, the power number to recirculation flow number ratio may sometimes be misleading and must be used with caution in selecting a mixing system.
Dual Impellers Dual impellers were examined in the 1-m CFT. The lower impeller was located at 0.2500 off-bottom clearance and the upper impeller 0.2500 below the liquid level. Table IX shows the principal power and flow parameters
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
Table IX. Principal Power and Flow Parameterr for the CFT Agitated by Dual 6IBTs and 3BSPPs and Their Single Components. Lower Impeller at 0.280D Off-Bottom Clearance and Upper Impeller 0.2800 below Liquid Level discharge induced recirculation discharge flow no. 1, flow no. 2, flow no., flow no. 2, impeller C e power no. Nqdt Nqd* Nqi Nqr2 GIBT sinele 0.250 1.68 GIBT single 0.250 1.67 2.91 GIBT dual 0.250 0.250 3BSPP single 0.250 0.331 0.60 0.49 1.09 3BSPP single 0.250 0.379 0.545 0.45 0.83O 0.42 1.2E1~ 3BSPP dual 0.250 0.250 a
Apparent numbers only, see text.
for this geometry, including some data for the individual propellers. The power numbers for the single GIBT at 0.2500 clearance and the GIBT located 0.2500 below the liquid level were found to be the same. For the dual GIBT arrangement, the resulting power number was 87% of the sum of the power numbers for single impellers. In contrast the 3BSPP located 0.2500 below the liquid level had 14.5% higher power number than the impeller located at 0.2500 off-bottom clearance. The power number for the dual arrangement was 76.7% of the sum of power numbers for single impellers. This result suggests that the 3BSPPs influence each other to a much higher degree than the 6IBTs. This was expected, because the discharge flow from the upper 6IBT is much wider and leas directed than from the 3BSPP and thus influences the lower impeller to a lesser degree. Comparison of the impeller discharge numbers for single and dual 3BSPPs revealed that the lower propeller discharge number is 1.38 times higher than that of the single propeller. One may ask why this increase? The answer is simple. In the definition of the discharge number, eq 4, the total volume of the tank is employed, while in the case of the lower impeller, part of the incoming flow is bypassing the upper strata of the vessel and therefore violating the very definition of the discharge number. The bypass loop circulates in a much spaller volume than the total tank volume, resulting in a higher number of passes and thus a higher discharge number for the lower impeller. Therefore the measured discharge number for the lower impeller can only be taken as an apparent discharge number. While there can be a value assigned to the ratio of the bypass flow to the fully circulating flow, there is no exact way to determine a mean fraction of the tank volume circulated by the bypass loop. Therefore the true discharge number of the lower impeller in dual impeller systems cannot be reliably calculated from the data obtained by the tracer method. Conclusions Relatively little difference was found in power numbers for the GIBT in the PBT and FBT, suggesting that the central profile of the PBT had little effect on the relatively wide discharge flow from the 6IBT. In contrast, the power numbers for the 3BSPP at 0.250 impeller clearance in the CFT and PBT were significantly higher than that for the FBT. This rise in power number was attributed to the restriction of the flow (head increase) caused by the central cone in the CFT and the central profile of the PBT. Minor geometric dissimilarities and errors in torque measurement were identified as possible causes for the discrepancies in power number data reported by various workers. The 3BSPP was shown to be inadequately described when only the propeller diameter, blade width, and propeller pitch were reported as parameters. The projected blade area and propeller blade height to width ratio were
found to strongly influence the magnitude of the power number. The discharge flow numbers for the 3BSPP were found to be higher in the PBT than in the FBT. Maximum impeller discharge flow numbers occurred for the GIBT at 0.2500 clearance and for the BBSPP at 0.3330 clearance. The maximums of the impeller induced flow numbers coincided with the maximums of impeller discharge flow numbers. The highest impeller recirculation flow number for all geometries agitated by the 3BSPP occurred in the CFT. The 3BSPP was shown to provide discharge flow and recirculation flow much more efficiently than the 6IBT. The power number to recirculation flow number ratios at 0.250 and 0.3330 impeller clearances were practically the same for all geometries agitated by the 3BSPP. However, equal ratios for these geometries do not mean that they would perform equally for any mixing task controlled by the recirculation flow. Therefore these ratios may sometimes be misleadipg and must be used with caution. Analysis of impeller power numbers for dual impellers in the CFT suggested a higher degree of interaction between dual 3BSPP's than between dual 6IBTs. It was also shown that the tracer method is unsuitable for determining the discharge number of the lower impeller.
Acknowledgmeqt I wish to express my thanks to M.D. Research Co. Pty.
Ltd.for permission to publish this paper, to J. M. Boyes and S. H. Marshall for the critical review of this paper, and to C. J. Goddard and P. H. Fuhry for their patience and care with the data collection.
Nomenclature Ab = projected blade area, m2 A, = blade camber area (Figure 41, m 2 a = baffle thickness, m b = baffle width, m C, = camber coefficient c = impeller clearance, m D = tank diameter, m d = impeller diameter, m d h = impeller hub diameter, m dl = central cone diameter, m d2 = peripheral fillet inner diameter, m e = upper impeller depth, m f = baffle Clearance, m g = gr8vitational acceleration, m h = impeller blade height, m h, = central cone height, m h2 = peripheral fillet height, m h, = central profie height, m 1 = blade length, m N = impeller speed, N p = power nurpber for homogeneous liquid (eq 1) N g d = discharge flow number (eq 3)
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 867
Nqi = induced flow number Nqr = recirculation flow number N14 = number of tracer passes n1-2= number of blades P = power (net shaft input), W p = propeller pitch Re = Reynolds number q d = discharge flow from impeller, ma s-l R, = profiled bottom radius, m = central profile radius, m = tank height, m t = time, s V = tank volume, m3 u = exponent (eq 1) w = impeller blade width, m x A = fraction of projected blade area xl+ = exponent (eq 1) TJ = dynamic viscosity, Ns m-2 p = liquid density, kg m-3
2
Subscripts d = dual impeller s = single impeller 1 = upper impeller 2 = lower impeller Abbreviations CFT = cone and fillet tank
FBT = flat bottom tank PBT = profiled bottom tank 3BSPP = 3-blade-square pitch propeller 6IBT = 6-inclined-bladeturbine ( 4 5 O ) Literature Cited Bates, R. L.; Fondy, P. L.; Corpateln, R. R. I d . Eng. Chem. Process Des. Dev. 1063, 2 , 310. Chudacek, M. W. "Proceedings of the 8th International Technical Conference on Slurry Transportation", Sakkestad, B.A., Ed.; San Francisco, Ca,March 15-18, 1983, p 185. Chudacek, M. W. Chem. Eng., in press. Fort. I.; Sedlakova, V. Collect. Czech. Chem. Commun. lBS8, 3 3 , 836. Hixson, A. W.; Luedeke, V. C. Ind. Eng. Chem. 1037, 2 9 , 927. Marr, 0. R.; Johnson, E. F. AICMJ 1063, 9 , 383. Medek, J. Chem. M m . 1078, 2 8 , 608. Medek, J.; Fort, I . Chem. M m . 1075, 2 9 , 62. Nagata, S. "Mixing - Prlnciples and Applications"; Kodansha Ltd.. Tokyo; Wiley: New York. 1975, p 125. Pwceiii, J. V.; Marr, 0. R. Jr.; Ind. En$. Chem. Fundem. 1062, 3 , 172. Rushton, J. H.; Costich, E. W.; Evert, H. J. Chem. Eng. Rog. 1050, 46, 395, 467.
Seichter, P. Chem. M m . 1075. 2 5 , 117. Stem. H. Chem. Llsty 1058, 5 2 , 839. Van de Vusse, J. G. Chem. €178.Sci. 1055, 4 , 178, 209. White, A. M.; Brenner, E. Trans. Am. Inst. Chem. Eng. 1034a, 30. 585. White, A. M.; Brenner, E.; Phillips, G. A.; Morrison, M. S. Trans. Am. Inst. Chem. Eng. 1934b, 30, 570. White, A. M.; Somerfwd, S. D. Chem. Metall. En$. 1936. 43, 370.
Received for review March 27, 1984 Revised manuscript received September 20, 1984 Accepted October 19, 1984
