Power Consumption in Stirred Tanks Provided with Multiple Pitched

bines.1 The present study extends that work to pitched- blade turbines (PBTs) and quantifies the total power requirements in mixing tanks provided wit...
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Ind. Eng. Chem. Res. 1999, 38, 2809-2816

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Power Consumption in Stirred Tanks Provided with Multiple Pitched-Blade Turbines Piero M. Armenante,*,† Barbara Mazzarotta,‡ and Gwo-Ming Chang† Department of Chemical Engineering, Chemistry and Environmental Science, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102-1982, and Department of Chemical Engineering, Universita` degli Studi di Roma “La Sapienza” Via Eudossiana 18, 00184 Roma, Italy

The power consumed by one, two, or three downward pumping, 45° six-blade pitched-blade turbines (6-PBTs) mounted on the same shaft was experimentally determined in stirred tanks under turbulent conditions. The power drawn by each individual impeller (in single- or multipleimpeller configurations), as well as the total power consumption, was measured with strain gauges mounted on the shaft and reported as individual or total power numbers. The power dissipated by single downward pumping, 45° four-blade pitched-blade turbines (4-PBTs) was also determined. Different combinations of the number of impellers, impeller diameter-to-tank diameter ratio, off-bottom clearance of the lowest impeller, Cb1, and spacing among impellers were tested. The overall power numbers of double 6-PBT systems were typically found to be smaller than twice the power number of a single 6-PBT. When Cb1 was low and the upper impeller distant from the lower impeller, the power number of the lower impeller and the overall power number were higher. Reducing the distance between the impellers lowered the overall power number and that of the lower impeller. Similar results were obtained with three 6-PBTs. The results of this work can be used to predict the power consumed by individual impellers in multiple 6-PBT systems in a turbulent flow regime and to determine the optimal impeller configurations to minimize energy consumption. Introduction The determination of the power consumed by rotating impellers in mixing tanks or reactors is important not only for the calculation of the energy requirements of the system but also for scaling up processes (such as those involving mass-transfer operations) in which power consumption per unit mass is a key parameter. Power consumption is a function of a number of parameters such as the type and number of impellers, stirring speed, physical properties of the fluid and the phases to be dispersed, and geometry of the system, including all dimensions and position(s) of the impeller(s) within the tank. Data are available in the literature on the power consumed by single impellers, but limited and scattered information is available on multiple-impeller systems. Even less information is available on the power dissipated by individual impellers in multiple-impeller configurations. Consequently, the designer often estimates the total power dissipated by multiple impellers from the knowledge of that consumed by single impellers. This is often incorrect. Recently, this research group has published data on the power dissipated by single and multiple disk turbines.1 The present study extends that work to pitchedblade turbines (PBTs) and quantifies the total power requirements in mixing tanks provided with one or more PBTs, as well as the power consumption of individual PBTs in multiple PBT systems. This was achieved * To whom correspondence should be addressed. Phone: (973)596-3548.Fax: (973)596-8436.E-mail: armenant@megahertz. njit.edu. † New Jersey Institute of Technology. ‡ Universita ` degli Studi di Roma “La Sapienza”.

experimentally for different combinations of the number of impellers, off-bottom clearance of the lowest impeller, and spacing among impellers. The results of this study can be used to predict the power requirements of single and multiple PBTs, as well as the power dissipated by individual PBTs in multiple PBT systems. This information can be important especially in all those cases in which the presence of additional impellers may be required to achieve not only a primary process requirement, such as solid suspension off the tank bottom, but also a secondary objective, such as a greater level of homogenization of the tank content (especially if the tank is tall). For example, Armenante and Uehara Nagamine2 and Armenante et al.3 have shown that the minimum agitation speed for complete off-bottom solid suspension (Njs) is fairly insensitive to the number of impellers but that the presence of a second impeller makes the flow near the tank bottom more parallel to it and reduces the fillet region near the tank wall. However, the power consumed by each impeller in the multiple-impeller systems was shown to be significantly lower (as little as 48% and as high as 84%) with respect to the single-impeller case and strongly dependent on the position of the lower impeller relative to the tank bottom. The resulting total power consumption also varied greatly, and for some impeller configurations (especially at very low impeller clearance), the total power consumed by two PBTs at Njs was nearly identical (as little as 7% different) with that of a single PBT also at Njs.2 Therefore, knowledge of the power consumed by individual impellers can help the designer decide the optimal number and location of impellers to achieve all the required process objectives while minimizing power consumption.

10.1021/ie980692o CCC: $18.00 © 1999 American Chemical Society Published on Web 06/09/1999

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Literature Review Single PBT Systems. Many investigators4-6 have experimentally determined the power characteristics of single PBTs and have found that the impeller power number, Ne, reaches a constant value for a given geometry if the agitation intensity is high enough to produce turbulent flow (Re > 10 000). Power numbers for single PBTs have been reported in a number of studies. Rushton et al.6 found that Ne for downward pumping pitched-blade turbines with six blades inclined at 45° (6-PBTs) was 1.42 when D/T ) 0.3, H/T ) 0.92, C′1/D ) 1, L/W ) 2.3, and B/T ) 1/10. Bates et al.4 produced Ne-Re plots for D/T ) 1/3, C′1/T ) 1/3, H/T ) 1. They also studied the effect of geometric factors such as blade angle and off-bottom clearance on the power consumed by 6-PBT impellers. O’Kane7 investigated the effect of blade width and the number of blades. The power number he obtained at standard conditions for a 6-PBT was 1.52. Shiue and Wong8 found Ne to be 1.74 for a downward pumping pitched-blade turbine having four blades inclined at 45° (4-PBT) in a tank with a hemispherical bottom (D/T ) 0.325, H/T ) 1, C′1/T ) 0.5, L/D ) 0.346, and wb/D ) 0.231). Chudacek9 reported a value for Ne of 1.63 for 6-PBTs in a flat-bottom tank (D/T ) 1/3, H/T ) 1, C′1/T ) 1/3, and W/D ) 0.2). Machon et al.10 found that Ne ) 1.72 for a 6-PBT with D/T ) 0.5, H/T ) 1, C′1/D ) 0.5, and an unspecified blade width. Raghava Rao and Joshi11 studied the flow pattern and power consumption in tanks stirred by different impellers. For downward pumping 6-PBTs (D/T ) 1/ , H/T ) 1, and w /D ) 1/ ), they found that Ne was 3 b 5 1.29, 1.35, and 1.61, when C′1 was equal to T/3, T/4, and T/6, respectively. Rewatkar et al.12 studied the effect on Ne of a number of geometrical factors such as impeller clearance, impeller diameter, blade angle, blade width, and number of blades. They found Ne to be 1.67 for a standard 6-PBT (D/T ) 1/3, C′1/D ) 1, and wb/D ) 1/5). Ne was strongly dependent on the flow pattern generated by the impeller and increased with decreasing offbottom clearance. They obtained the following overall dimensional correlation for the power number of a 6-PBT:

() ( )

Ne ) 0.653T0.26

T D

0.11

C′1 T

-0.23

nb0.68A1.82

(1)

where T is in meters and the equation is valid for 6 e T/D < 3, W/D ) 0.3, H/T ) 1, 0.125 e C′1/T e 0.33, 0.5 e A e 1.05, and 4 e nb e 8. Fasano et al.13 produced Ne-Re plots for 4-PBTs with D/T as the parameter. Armenante and Uehara Nagamine14 obtained power numbers for 6-PBTs as a function of D/T and Cb1/T and showed that Ne decreased with impeller clearance in the Cb1/T range tested (1/48-1/4). Multiple PBT Systems. Little information is available on multiple PBT systems. Bates et al.4 studied the effect of impeller spacing on power consumption in twoimpeller systems. They found that a dual 6-PBT system does not consume twice the power of a single turbine for S12/D e 4 and that the total power dissipated decreases with impeller spacing for S12/D < 1. Other investigators have measured the power dissipated by two 6-PBTs pumping against each other, i.e., one upward and the other downward.15,16 Because of the different fluid dynamic regimes, this configuration cannot be directly compared to that studied here. Armenante and Uehara Nagamine2 studied solid sus-

Figure 1. Setup for experiments with 6-PBTs: (a) experimental apparatus; (b) tanks used in this work [Tank #1 (H/T ) 1; B/T ) 0.1); Tank #2 (H/T ) 2; B/T ) 0.088)]. Table 1. Dimensions of Tanks Used in the Experiments with 6-PBTs

tank

tank diameter, T (m)

tank height (m)

liquid height, H (m)

#1 #2

0.289 0.289

0.386 0.688

0.289 0.578

H/T

baffle width, B (m)

B/T (%)

1 2

0.029 0.025

10 8.8

pension and power consumption in single and double 6-PBT systems when the lower impeller was close to the tank bottom (1/48 < Cb1/T < 1/8). They found that the lower and upper impeller had different power numbers and that the total power consumed was smaller than twice that consumed by a single impeller. No additional information is available on the power dissipated by the individual PBTs in multiple-impeller configurations. Triple PBT systems have not been studied. Apparatus and Method Experiments with 6-PBTs. Figure 1a shows a diagram of the apparatus. Agitation was provided by a 2.0 HP variable-speed motor (G. K. Heller Corp., Floral Park, NY) with a maximum speed of 1800 rpm. The rotational speed was measured with a digital tachometer connected to a photoelectric pick-up sensor (ColeParmer, Chicago, IL), accurate to within (1 rpm. The tanks consisted of two open, flat-bottomed, cylindrical, Plexiglas vessels provided with four baffles spaced 90° apart and filled to different liquid heights. Their geometric characteristics are given in Figure 1b and Table 1. The impellers were downward pumping 6-PBTs with a diameter of 0.0762 m. The blades were inclined at a 45° angle. The blade width projected along the vertical axis, wb, was equal to D/8 (9.53 mm, i.e., W ) 13.5 mm). The length of each blade was 27 mm, and the blade metal thickness, k, was 1.6 mm. Depending on the experiment, one, two, or three impellers were mounted

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on the shaft. When the effect of the impeller diameter on power consumption was studied, another geometrically similar impeller with a diameter of 0.102 m was also used. The shafts were hollow aluminum tubes of different lengths provided with strain gauges (Measurements Group Co., Raleigh, NC, Part No. CEA-06-187UV-350) mounted above the impellers. The gauges were electrically connected to a signal conditioner and an amplifier system (2120A system, Measurement Group Co., Raleigh, NC) via a slip ring assembly (Airflyte Electronics Co., Bayonne, NJ, Part No. CAY1030-12-2), so that the torque and hence the power dissipated by each impeller could be experimentally determined. The data were collected and analyzed with a data acquisition system (Labtech Notebook) connected to a computer. The sampling frequency of the data acquisition system was 30 min-1. The value of each experimental variable was determined by calculating the average of the 30 readings obtained during a 60 s sampling time. The reproducibility was within (5.2%. Additional details on apparatus construction, operation, and calibration and data analysis are given elsewhere.1 The distances, Sij, between the impellers could be varied, and the tank could be translated vertically to change the position of the impeller-shaft assembly relative to the tank (and especially the off-bottom impeller clearance of the lowest impeller, Cb1, measured from the bottom of the impeller blade). Single-impeller experiments were carried out in Tanks #1 and #2 (Figure 1b). Double- and triple-impeller runs were conducted, respectively, in Tanks #1 and #2. In the latter case the spacing between the top and bottom impellers (S13) was kept constant, while the position of the middle impeller was varied. All experiments were performed at room temperature using tap water. Its physical properties were taken at 19 °C.17 Once the power consumed by each impeller, Pi, was experimentally derived, the corresponding Newton number (power number) for each impeller, Nei, was calculated as6

Nei ) Pi/FN3D5

(2)

The impeller Reynolds number was defined as Re ) FND2/µ. All experiments were run in duplicate at agitation speeds of 6.67 and 8.33 rps (400 and 500 rpm, respectively), corresponding to Re in the range of 38 700-48 400 (fully turbulent regime). The results were interpreted using the following equation:1

Nei ) f

(

)

Cb1 T S12 S23 , , , ,n D D D D

(3)

The assumptions under which this equation holds are discussed elsewhere.1 Experiments with 4-PBTs. Agitation was provided by a variable-speed motor (300-2000 rpm) connected to a 15-mm-diameter shaft and provided with a stroboscopic optical sensor with digital readout (IKA Labortechnik, Staufen, Germany) to measure the agitation speed (reproducibility: (1 rpm). The agitation tank was a glass vessel, having an internal diameter of 0.3 m and a height of 0.445 m. The tank was provided with four metal baffles, 30 mm wide, 5 mm thick, and spaced 6 mm away from the tank wall. The gap between the baffles and the tank bottom was 9 mm. The tank was filled with tap water up to a height equal to the tank diameter. The impellers were downward pumping 4-PBTs

Figure 2. Single-impeller system: (a) effect of the Cb1/D ratio on the power number; (b) effect of the D/T ratio on the power number.

having four blades inclined at 45° and diameters of 0.103 m (L ) 41.5 mm, W ) 20 mm, k ) 4 mm, hub diameter ) 20 mm, and hub height ) 27 mm) and 0.152 m (L ) 66 mm, W ) 30 mm, k ) 4 mm, hub diameter ) 20 mm, and hub height ) 27 mm), respectively. Only one impeller at a time was mounted on the shaft. Shafts of different lengths were used so that the impeller clearance could be varied in the range of 11-222 mm. The torque was measured with a commercial torquemeter (Leane International, Parma, Italy; model TR10C) consisting of a short shaft provided with four strain gages and a brush-free electronic signal transmitter connected to a signal conditioner and digital readout meter (Digitec Corp., Marion, OH; model IQ-280A). The reproducibility of the measurement was 0.3 N‚m. Experiments were conducted at 6.67 and 8.33 rps (400 and 500 rpm, respectively) using the 0.103 m impeller and at 5 and 6.67 rps (300 and 400 rpm, respectively) using the 0.152 m impeller, corresponding to Re in the range of 69 300-154 100 (fully turbulent regime). The results were interpreted as mentioned above. Additional information is provided elsewhere.18 Results and Discussion Single 6-PBT and 4-PBT Systems. Ne for 6-PBTs was found to be independent of N (standard deviation: 1.76%) when the flow was turbulent. Regressing P vs N resulted in an exponent of N equal to 2.91 ( 0.03 (Tank #1; D/T ) 0.264; Cb1/D ) 1; H/T ) 1), i.e., very close to the value of 3 that can be obtained from eq 2 when Ne is constant.4 Figure 2a shows the effect of off-bottom clearance on Ne for single PBTs. The power number for 6-PBTs is within the range of 1.39-1.9 and is about 1.5-1.6 for Cb1/D g 3. A minimum can be observed for Cb1/D ) 1.5.

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Figure 3. Double 6-PBT system: power numbers of individual impellers and total power number (equal to the sum of power numbers of individual impellers) as a function of (a) Cb1/D ratio and (b) S/D ratio. Note: NeSingle Std. ()1.49) was obtained at Cb1/D ) 1, as per its definition, and is not a function of Cb1/D.

For lower Cb1/D values Ne was found to increase with decreasing impeller clearances. As observed before,11,14 this phenomenon can be attributed to the change in flow direction that occurs when the downward jet discharged by the impeller impinges on the tank bottom (“throttling effect”). The smaller the impeller clearance, the more abrupt the change in flow direction will be (from downward to upward), generating more turbulence and resulting in an increased power consumption. For Cb1/D ) 1, D/T ) 0.264, and H/T ) 1 (taken here as the “standard conditions”), Ne for 6-PBTs was found to be 1.49 (NeSingle Std.). This value is in line with those reported by previous investigators under similar conditions (1.42, Rushton et al.;6 1.3, Bates et al.;4 1.52, O’Kane;7 1.7 (wb/D ) 1/5), Kuboi and Nienow;16 1.63, Chudacek;9 1.72, Machon et al.;10 1.29 (wb/D ) 1/5), Raghava Rao and Joshi;11 1.67 (wb/D ) 1/5), Rewatkar et al.12). The dependence of Ne on Cb1/D found here (Figure 2a) is in agreement with that of Bates et al.,4 who examined a range for Cb1/D (Cb1/D < 1.2) narrower than that studied here and could only report a monotonic decrease of Ne with increasing Cb1/D. Figure 2a shows that the dependence of Ne on Cb1/D for 6-PBTs is more complex and that a minimum in Ne occurs for Cb1/D in the range of 1.5-2. Chudacek9 also reported Ne to decrease with increasing Cb/T: for a 6-PBT with W/D ) 0.2, Ne was 1.63, 1,75, 1.92, and 2.07 for Cb/T equal to 1/3, 1/4, 1/6, and 1/12, respectively. The ratios among these Ne values are very similar to those found here. Raghava Rao and Joshi11 also reported increasing Ne’s for decreasing impeller clearances. For example, when C′1 was equal to T/6, the power number they found was 24% greater than that at C′1 ) T/3, while in this work the corresponding increase in Ne was 17%. In general, the Ne values they obtained were lower than

Figure 4. Double 6-PBT system: (a) effect of the Cb1/D ratio on the ratio of the total power number (equal to the sum of the power numbers of individual impellers) to the power number of a single impeller at standard conditions (Cb1/D ) 1, NeSingle Std. ) 1.49); (b) effect of the Cb1/D ratio on the ratio of the power number of the upper impeller (Ne2) to that of the lower impeller (Ne1).

Figure 5. Double 6-PBT system: (a) effect of the S/D ratio on the ratio of the total power number (equal to the sum of the power numbers of individual impellers) to the power number of a single impeller at standard conditions (Cb1/D ) 1, NeSingle Std. ) 1.49); (b) effect of the S/D ratio on the ratio of the power number of the upper impeller (Ne2) to that of the lower impeller (Ne1).

those found here despite their use of larger impellers. More recently, the same group12 reported higher values

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Figure 6. Triple 6-PBT system: (a) power numbers of individual impellers and the total power number (equal to the sum of the power numbers of individual impellers) as a function of the Cb1/D ratio; (b) effect of the Cb1/D ratio on the ratio of the total power number to the power number of a single impeller at standard conditions (Cb1/D ) 1, NeSingle Std. ) 1.49).

for Ne, which can also be calculated with eq 1. For example, for D/T ) 0.264, C′1/D = Cb/D ) 1, and nb ) 6, A ) 45° (the “standard conditions” in this work), eq 1 predicts Ne to be 1.62. This value can be corrected by extrapolating the data in Rewatkar et al.’s Figure 912 to account for the different W/D ratio used here (0.177). The resulting value for Ne is approximately 1.5, i.e., very close to the value found here (1.49). Rewatkar et al.12 also reported an increase in Ne with decreasing impeller clearance (e.g., 20% more when C′1/T was 1/6 than when was it was 1/3, compared to a corresponding increase of 17% in this work). Figure 2a also shows the results for single 4-PBTs. For D/T ) 0.343 the dependence of Ne on Cb1/D for a 4-PBT was found to be very similar to that of 6-PBTs. However, for D/T ) 0.507 the dependence was very different, especially in a narrow range of Cb1/D (0.30.6) where a minimum was observed. Additional experiments (as well as control and calibration experiments) conducted at a later date to verify the data confirmed the results obtained previously. A possible explanation for these results is that in the region in which the experiments with the larger impeller were conducted the impeller may experience flow reversal.19 Accordingly, the downward-angled discharge flow generated by the PBT impinges on the tank wall rather than the tank bottom, resulting in a different power number. The Ne value found here (1.46) for Cb/D ) 1 is in line with the value of 1.74 of Shiue and Wong8 for a larger impeller (W/D ) 0.326 instead of 0.194 in this work) and the power numbers (∼1.2-1.4) of Fasano et al.13 The effect of Cb1/D on Ne for 6-PBTs and 4-PBTs differs significantly from that for disk turbines, for which Ne increases monotonically with Cb1/D and approaches an asymptotic value for large Cb1/D values.1

Figure 7. Triple 6-PBT system: ratio of power consumed by each impeller to the total power consumption as a function of Cb1/D: (a) S12/D ) 0.667; (b) S12/D ) 2.5; (c) S12/D ) 4.

The liquid height has a negligible effect on power consumption (Figure 2a), if no surface aeration or air entrainment is present.6,12 The extent of baffling (B/T ratio equal to 10% and 8.8% for Tank #1 and #2, respectively) has a minor effect on Ne once fully baffled conditions are achieved. In Figure 2b Ne is plotted as a function of D/T, keeping either Cb1/D or Cb1/T constant. The effect of the D/T ratio on Ne for 6-PBTs is very limited in both cases. Ne varied more appreciably with D/T for 4-PBTs, also because the D/T range was higher. Previous investigators13 also reported that Ne dropped with D/T although not monotonically, because large impellers (D/T ) 0.5) had power numbers much larger than those of smaller impellers. However, a comparison between parts a and b of Figure 2 shows that the difference in Ne between 4-PBT impellers with different D/T ratios increases more sharply when 0.3 < Cb1/T < 0.6. Double 6-PBT Systems. The effect of Cb1 on the power numbers of individual impellers is shown in Figure 3a. This figure also shows the value of NeSingle Std. (obtained, by definition, at Cb1/D ) 1 and therefore independent of Cb/D) as a reference horizontal line. Neither Ne1 nor Ne2 changes appreciably with Cb1/D except for Ne2 at low Cb1/D values. Comparing Figures 2a and 3a shows that while Ne1 is similar, for all Cb1/ D’s, to NeSingle Std., Ne for single impellers increases with decreasing Cb1/D. This implies that the upper impeller exerts a dampening effect on the power number of the lower impeller at low Cb1/D values. Despite the appreciable impeller spacing (S/D ) 1.5), Ne2 is always smaller than Ne1, probably as a result of the overall

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Figure 8. Triple 6-PBT system: (a) power numbers of individual impellers and the total power number (equal to the sum of the power numbers of individual impellers) as a function of the S12/D ratio; (b) effect of the S12/D ratio on the ratio of the total power number to the power number of a single impeller at standard conditions (Cb1/D ) 1, NeSingle Std. ) 1.49).

circulation flow and the fact that the upper impeller flow is less affected by the presence of the tank bottom than the lower impeller. This behavior is opposite to that of radial impellers (disk turbines), for which Ne1 is always smaller that Ne2.1 The effect of S is shown in Figure 3b. Both impellers always have individual power numbers smaller that NeSingle Std., although Ntot stays nearly constant with S. For S/D g 1 the upper impeller always dissipates less power than the lower impeller. Only when the impellers are closer together (S/D ) 0.667) does the situation reverse itself. The effect of Cb1 on power consumption as a function of S/D is reported in Figure 4a,b. Combined, these parts can be used to calculate the individual and total turbulent power numbers as a function of Cb1/D for any double 6-PBT system for which H/T ) 1 and the impellers are geometrically similar to those used here. Double 6-PBT systems consume significantly less power (12-25%) than two single 6-PBTs, even when the impellers are well spaced (Figure 4a). Only for large spacing (S/D ) 1.5) and low impeller clearances (Cb1/D ) 0.333) does Ntot become equal to twice NeSingle Std. This implies that the addition of a “tickler impeller” (i.e., an impeller close to the tank bottom) to another impeller placed higher in the tank may be advantageous from a process point of view (if agitation is needed even when the tank is nearly empty) but not from an energy standpoint. For S/D g 1 the upper impeller almost always consumes less power than the lower impeller (Figure 4b). The opposite is true, somewhat unexpectedly, for smaller spacing (S/D ) 0.667). One can speculate that under such conditions the proximity of the two impellers

reduces the throttling effect described above, lowering the power consumption of the lower impeller as well as the overall power consumption. A comparison between parts a and b of Figure 4 for 6-PBTs and the corresponding figures for disk turbines1 reveals that the behavior of these two types of impellers is very different when it comes to the dependence of Nei on impeller clearance and spacing: reducing Cb1/D lowers the power consumption of disk turbines but increases that of 6-PBTs. In Figure 5a Netot/NeSingle Std. is plotted as a function of S/D with Cb1/D as the parameter. These data show again that the power dissipated by two 6-PBTs is typically 10-20% lower than twice that of a single 6-PBT, except when the lower impeller is very close to the tank bottom (Cb1/D ) 0.333) and the spacing is in a narrow range (1.33 < S/D < 2). In general, the lower impeller draws more power than the upper one, especially at larger spacing (Figure 5b) but not at low S/D ratios (S/D ) 0.667). Bates et al.4 also reported that dual 6-PBT systems consume less power than twice the power of a single 6-PBT. However, they only examined one clearance. Triple 6-PBT Systems. In Figure 6a, individual (for each impeller) and cumulative (total) Ne values are plotted as a function of Cb1/D. As before, this figure also shows NeSingle Std. (Cb1/D ) 1) as a reference horizontal line. For 0.5 e Cb1/D e 1.5 all three impellers have similar power consumptions, and all Nei’s are slightly smaller than NeSingle Std., probably because each impeller reinforces the rotating flow generated by the others. When the off-bottom clearance of the lower impeller is reduced, Ne1 increases, similarly to the single- and double-impeller cases (Figures 2a and 4a, respectively), Ne2 decreases by about the same amount, and Ne3 remains nearly unchanged. The net result is that Netot changes only slightly (less than 7%) even when Cb1/D is varied in the entire range tested here. The effect of off-bottom clearance on Netot is shown in Figure 6b. The total power consumption is 4-18% smaller than 3NeSingle Std. and decreases nearly monotonically with Cb1/D. Figure 7 shows the effect of Cb1/D on the fractional power consumption of each individual impeller, Pi/Ptot. When the lower impeller is very close to the middle impeller (S12/D ) 0.667), the latter typically consumes more power than the former (Figure 7a). This is the reverse of what is observed at higher S12/D ratios (Figure 7b,c) but is consistent with the double-impeller case for S12/D ) 0.667 (Figure 4b). The lower impeller typically consumes significantly more power than the other two if it is near the tank bottom (Figure 7b,c) or the upper and middle impellers are close together (Figure 7c). Figure 8a shows the variation of Netot and Nei as a function of S12/D, for Cb1/D ) 1. The total power number remains nearly unchanged independently of S12/D (maximum deviation: 4.9%). However, Ne1 is larger or smaller than Ne2 depending on the value of S12/D. In Figure 8b the effect of spacing between the lower and middle impellers on Netot is reported at different Cb/D’s. Full power is drawn only when the lower impeller is very close to the tank bottom, as observed before, and only for some S12/D values. In all other cases Netot is lower than 3NeSingle Std. by as much as 15%. Confirming the conclusions reached with the two-impeller system, the presence of a “tickler impeller” significantly in-

Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2815

impeller distant from the lower impeller, the throttling effect caused an increase in the power number of the lower impeller and hence the overall power number. Reducing the distance between the impellers limited this effect and lowered the overall power number as well as the power number of the lower impeller. In triple 6-PBT systems with equally spaced impellers, the cumulative power dissipated was typically lower than that of three single 6-PBTs and relatively constant over a wide range of Cb1/D values. However, at low Cb1/D the lower impeller power consumption increased and that of the middle impeller decreased. The results obtained in this work can be used to determine the overall power consumption as well as the power consumed by individual 6-PBTs in single- and multiple-impeller systems under turbulent flow conditions. Acknowledgment This work was partially supported by the National Science Foundation (Grant No. EEC 9520573), whose contribution is gratefully acknowledged. List of Symbols

Figure 9. Triple 6-PBT system: ratio of the power consumed by each impeller to the total power consumption as a function of S12/ D: (a) Cb1/D ) 0.167; (b) Cb1/D ) 0.5; (c) Cb1/D ) 1.

creases the overall power consumption of the system with respect to configurations in which the third impeller is placed not as close to the tank bottom. However, even in such a case, the overall energy consumption is only as high as that of three “standard” single impellers. Finally, in Figure 9 the effect of impeller spacing on the individual fractional power consumptions is given. At low impeller clearance (Cb1/D ) 0.167), the throttling effect is clearly present and the power dissipated by the lower impeller is always higher than that of the upper impeller (Figure 9a). P1 is also larger than P2 except when the lower and middle impellers are close to each other (S12/D e 1). A similar trend, although not as pronounced, can be seen at a higher Cb1/D value (0.5; Figure 9b). However, for Cb1/D ) 1 (Figure 9c) the lower impeller typically consumes less power than the upper one (but not for S12/D > 3). These results indicate that except at low Cb1/D’s the relative power consumptions of 6-PBTs in a triple-impeller configuration are relatively similar. This is the opposite of what is observed with disk turbines.1 Conclusions The power numbers of single PBTs were found to be relatively constant in a wide Cb1/D range and to increase with decreasing Cb1 for Cb1/D < 1 because of the throttling effect caused by the proximity of the impeller to the tank bottom. The overall power number of double 6-PBT systems was typically found to be significantly smaller than twice the power number of a single 6-PBT. However, when the Cb1/D ratio was low and the upper

A ) impeller blade angle (rad) B ) baffle width (m) Cb1 ) off-bottom clearance of the lower impeller, measured from the bottom edge of the impeller blade to the bottom of the tank, as shown in Figure 1b (m) Cb2 ) off-bottom clearance of the upper impeller (in doubleimpeller systems) or the middle impeller (in tripleimpeller systems), measured from the bottom edge of the impeller blade to the bottom of the tank, as shown in Figure 1b (m) Cb3 ) off-bottom clearance of the upper impeller (in tripleimpeller systems), measured from the bottom edge of the impeller blade to the bottom of the tank, as shown in Figure 1b (m) C′1 ) off-bottom clearance of the lower impeller, measured from the middle of the impeller to the bottom of the tank (m) D ) impeller diameter (m) H ) height of liquid in the tank (m) L ) blade length (m) n ) number of impellers nb ) number of blades N ) agitation speed (rotations/s, rps, or rotations/min, rpm, as indicated) Ne ) Newton number or power number Nei ) Newton number or power number of individual impeller i (with i ) 1, 2, or 3), as defined in eq 2 Netot ) total (cumulative) Newton number or power number of all impellers NeSingle Std. ) Newton number or power number of a single impeller at standard conditions (Cb1/D ) 1, D/T ) 0.264, and H/T ) 1) Pi ) power drawn by individual impeller i (with i ) 1, 2, or 3; W) Ptot ) total (cumulative) power drawn by all impellers (W) PSingle Std. ) power drawn by a single impeller at standard conditions (Cb1/D ) 1, D/T ) 0.264, and H/T ) 1; W) Re ) impeller Reynolds number (FND2/µ) S ) spacing (distance) between impellers in double-impeller systems (m) Sij ) spacing (distance) between impeller i and impeller j (with i ) 1, 2, or 3; j ) 2 or 3) in triple-impeller systems (m) T ) tank diameter (m)

2816 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 W ) blade width (m) wb ) blade width projected along the vertical axis (m) Greek Symbols F ) liquid density (kg/m3) µ ) liquid viscosity (kg/(m‚s)) Acronyms PBT ) 45° pitched-blade turbine 4-PBT ) four-blade 45° pitched-blade turbine 6-PBT ) six-blade 45° pitched-blade turbine

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Received for review November 5, 1998 Resubmitted for review April 20, 1999 Accepted April 28, 1999 IE980692O