Power Consumption in Gas-Inducing-Type Mechanically Agitated

+

+

Ind. Eng. Chem. Res. 1996, 35, 1583-1602

1583

PROCESS DESIGN AND CONTROL Power Consumption in Gas-Inducing-Type Mechanically Agitated Contactors Kaliannagounder Saravanan, Vishwas D. Mundale, Ashwin W. Patwardhan, and Jyeshtharaj B. Joshi* Department of Chemical Technology, University of Bombay, Matunga, Bombay 400 019, India

Power consumption was measured in 0.57, 1.0, and 1.5 m i.d. gas inducing type of mechanically agitated contactors (GIMAC) using single and multiple impellers. The ratio of impeller diameter to vessel diameter was varied in the range of 0.13 < D/T < 0.59. The effect of liquid submergence from the top and impeller clearance from the vessel bottom was investigated in detail. In the case of multiple impeller systems, six different designs were investigated. The designs included pitched blade downflow turbine (PBTD), pitched blade upflow turbine (PBTU), downflow propeller (PD), upflow propeller (PU), straight bladed turbine (SBT) and disc turbine (DT). The effect of interimpeller clearance was studied for the multiple impeller system. The effect of impeller speed was studied in the range of 0.13 < N < 13.5 rotations/s. A mathematical model has been developed for power consumption before and after the onset of gas induction. 1. Introduction

2. Literature Survey

There are several industrially important unit processes such as hydrogenation, alkylation, ozonolysis, oxidation, hydrochlorination and hydrobromination, ammonolysis, addition halogenation, etc., where it is desirable to have practically complete utilization of the solute gas. In the conventional stirred reactors, the gas is sparged at the bottom and the gas phase is finely dispersed and uniformly distributed by the mechanical agitation. However, in a large number of cases, the gas phase conversion per pass is low, due to limited residence time of the gas and/or the low rate of chemical reaction. The gas phase utilization can be improved by using different reactor designs such as sparged loop reactors, surface aerators, and gas-inducing-type mechanically agitated contactors (GIMAC). The scope, classification, and comparison of GIMAC with other reactors have been reported by Saravanan et al. (1994). They have developed a model for the critical impeller speed for the onset of gas induction and the rate of gas induction for a single impeller system. Saravanan and Joshi (1995) have studied the rate of gas induction for a dual impeller system. The subject of power consumption in GIMAC for a single impeller system has been studied by few investigations [Arbiter (1969) and Zundelevich (1978)]. There is practically no information in the published literature regarding the power consumption in multiple impeller systems, with gas induction. The present work aims at developing (i) a systematic understanding of power consumption performance of an impeller/stator assembly. Both single and dual impeller systems have been considered (ii) a mathematical model for power consumption.

Arbiter et al. (1969) have reviewed the previous efforts of modeling the power consumption in gas-inducing flotation cells and developed a new model. The approach is briefly summarized below: (1) The density of the gas liquid dispersion was assumed to be an average of the gas and liquid densities weighed with total gas flow rate (induced gas plus recycled gas) and the total liquid flow rate generated by the impeller. The ratio of recycled gas rate to the gas induction rate was assumed to be practically constant. (2) The ratio of liquid flow generated by the impeller to the rate of gas induction was related to the gas flow number. This was achieved through an empirical relationship obtained through experiments on a centrifugal pump. The relationship is as follows:

* Author to whom correspondence may be addressed.

( )

Q1 1 )C Qg NQg

n

+d

(1)

(3) Impeller power number was assumed to be constant and equated to the power number in liquid alone. These assumptions led to the following equation relating the power consumption of gas inducing impellers:

Pg/PL 1 - (Pg/PL)

)

{( ) }

Fa 1 R FA NQg

n

+β

(2)

where R is the ratio of liquid flow to gas recirculation flow in the impeller zone. Constant β is indicative of the ratio of reduction in the liquid flow (due to gas induction) to the gas recycle flow in the impeller zone. Equation 2 has been very successful in correlating their

+

+

1584 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996

own data as well as the data from other investigators. However, the approach suffers from the following limitations: (1) The equation depends on the knowledge of power consumption in the absence of gas induction. These data are very difficult to obtain experimentally as the gas induction begins very quickly. Attempts to thwart the gas induction by blocking the gas entry are not very reliable as the hydrodynamics of the impeller may be affected in an unpredictable manner. (2) The equation also depends on the knowledge of the gas induction rate. Although this is not a handicap, errors associated with estimating the gas induction rate would add to the other errors of estimation. (3) The original empirical relation relating the liquid and air flow rates (see eq 1) is not completely established in respect of dependence on the impeller diameter. (4) Assumption of the constancy of impeller power number between gassed and ungassed impeller cannot be substantiated. (5) The equation has been tried only on the laboratory scale systems. Zundelevich (1978) has attempted simulation of the gas induction and power consumption by drawing an analogy to a liquid jet gas ejector. He made the following assumptions: (1) Power number of the impeller is constant even in the presence of gas induction. (2) Density of dispersion around the impeller is an average of the gas and liquid densities, weighted with the gas and liquid flow rates. By appropriate manipulations, an optimization parameter was obtained as

(

QgS CH N2Qg + NQg ) Pg NpFg η

)

According to Hixon and Baum (1942), the process of power consumption by the impeller is related to the torque acting on the impeller blades. The torque arises out of a drag force acting on the impeller blade at various locations, due to the relative velocity between the fluid and the impeller blade. The drag force on an element of blade can be related to the relative velocity between the fluid and the blade, the area of blade perpendicular to the flow and a drag coefficient. The following assumptions were made: (i) Density of fluid phase surrounding the impeller is assumed to be a constant. (ii) The vertical projected width of the impeller, W, is assumed to be a constant irrespective of the radial position. (iii) The actual relative velocity of the fluid and the impeller blade in the tangential direction is assumed to be proportional to the tangential velocity of the impeller blade through a constant slip factor, fs. Thus, the relative velocity of the fluid with respect to the impeller is equated to fs(2πrN) at any radial location. (iv) The drag force acting on the blades is assumed to be a steady state process represented by Newton’s law through a drag coefficient CD. (v) Any drag caused by radial or axial flows and an associated power consumption is assumed to be represented by an altered value of the slip factor fs and the drag coefficient CD. The above set of assumptions translates into the following set of equations.

dF ) CDF(U2/2) dA

(3)

where

(2A)

For closed impellers (type 1 of Saravanan et al. (1994)) the optimum value of QgS/Pg was 1.08 × 10-5 m4/W s for an air-water system. Behavior of the system around the optimum was represented by η ) 0.071 and Np ) 3.1 in eq 2A above. On the other hand for the turbo aerator design, which is capable of inducing a larger gas quantity, the optimization parameter QgS/ Pg was 0.85 × 10-5 m4/W s with η ) 0.048 and Np ) 2.5. The success obtained by Zundelevich in correlating their own data is remarkable. This implies the validity of the assumptions that underlie their analysis for their experiments. However, experiments by Raidoo et al. (1987) on the turbo aerator design at 0.15, 0.2, and 0.25 m diameter scale did not produce a unique relationship between the impeller head number CH and the gas phase Euler number, Eug. This suggests the necessity of additional experimental work particularly using large scale equipment. Further, there is a need to revisit the assumptions made in the previous work and to develop a new mathematical model. 3. Models for Power Consumption

dA ) W dr U ) 2πrNfs Combining the above equations

(

)

CDWF (2πfsN)2r2 dr 2

dF )

The torque caused by this force on the shaft is

dτ ) r dF )

(

)


(4)

Integrating between torque limits from 0 to τ over radius 0-R for the impeller

τ)

(

)

CDWF (2πfsN)2(R4/4) 2

(5)

The power consumption P will be

P ) 2πNτ ) (CDf 2s π3)(FWR4N3) For geometrically similar impellers, W ) RD 2 3

3.1. Power Consumption in Liquid Alone. The conventionally understood power number is essentially derivable from dimensional analysis. However, its interpretation as a measure of drag coefficient of the impeller blades (Hixon and Baum, 1942) has been considered here for its adaptability to a situation of gasinducing impeller.

RCDf s π P ) Np ) 3 5 16 FN D

(6)

Power number as understood in conventional analysis would remain constant when (a) impeller aspect ratio R is held constant, (b) flow is turbulent and impeller drag coefficient (in liquid alone) is constant, (c) slip ratio

+

+

Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1585

occurring between impeller and fluid is a constant, or (d) alternately, Cf 2s product remains constant. It is, however, convenient to define a modified power number as a rationalized or scaled drag coefficient, C*

P ρWR4N3 C*D ) CDfs2π3

N* p ) C* D ) where

}

(7)

3.2. Model for Power Consumption for GIMAC in the Absence of Gas. The discussion of hydrodynamics of the impeller zone (Saravanan et al., 1994) describes the need for a net inflow into the impeller zone, which is in equilibrium with the net outflow from the impeller zone. This zone of fluid recycle is found to be at the eye of the impeller. Such recycle of fluid could exert a torque on the impeller contributing to or reducing the power consumption. These considerations are translated to the following assumptions. (1) A constant torque τr is assumed to be acting on the impeller at all times, due to fluid recycled to the impeller. (2) The slip factor is assumed to be constant over the impeller. Although this is not fundamentally justifiable, it is practically acceptable. This arises out of our inability to separate the product Cf 2s in the current analysis. Integrating eq 4 within the range from 0 to R

∫ττ dτ ) ∫0R r

(

)


(8)

and simplifying

τ - τr )

(WF8 )(2πf N) (C R ) 2

s

4

D

(9)

For P ) 2πNτ

P ) τr(2πN) + (CDf 2s π3)(FWN3R4)

(10)

In the case of gas-inducing impellers as described by Saravanan et al. (1994), the impeller is covered by a stator which prevents the flow of liquid from above, hence all the liquid recycled to the impeller takes place from below. This liquid recycle is associated with a torque τr. If the stator is absent the torque τr ) 0, and in that case the above model boils down to the conventional model (eq 7). Since CD cannot be separated from the slip factors, a scaled drag coefficient can be defined as in eq 7. Equation 10, therefore, takes the following form: 3 4 P ) τr(2πN) + C* D(FWN R )

(11)

3.3. Power Consumption in the Presence of Gas Induction. In the conventional mechanically agitated contactors where gas is sparged below the impeller, the variation of power number with impeller speed is very well known (Tatterson, 1991). In this case four regions can be identified. In the first region, the power number increases with an increase in the speed. This occurs because the flow pattern generated by gas sparging is in a opposite direction to that generated by the impeller. Hence, a higher torque is required and the power number increases. As the impeller speed increases further the flow generated by the impeller continuously increases, and

it provides increasing resistance to the flow of gas. The gas holdup increases steadily, and the power number decreases. This can be identified as the second region. The impeller action is still not sufficient to deviate the flow behavior of the gas phase markedly, and most of the gas rises vertically along the shaft. Since bubble breakup does not occur by the impeller action in this region, the impeller is said to be flooded with the gas and this region is called the flooding region. The impeller rotation generates a low-pressure region behind the impeller blades. The extent of pressure reduction depends upon the kinetic head (π dN)2/2. At a certain impeller speed, the reduction in pressure is sufficient to hold the gas phase in the low-pressure region against the buoyant force. The gas accumulated behind the impeller is called the cavity. The size of the cavity formed increases as the extent of pressure reduction increases, i.e. as the impeller speed increases. Due to the cavity formation the intensity of eddy motion behind the impeller decreases, and the turbulent energy dissipation behind the impeller blade decreases. Therefore the power number decreases. As the impeller speed increases further the shear stress field generated by the impeller is strong enough to break the cavity. Just at this point the cavity size is at a maximum which corresponds to the minima in the power number. An increase in the impeller speed at this point causes an increase in the cavity breakup causing a reduction in the cavity size. As a result the power number goes on increasing. This is observed in the third region. The impeller action starts contributing to bubble size. In his region a certain critical speed is reached when the gas bubbles even go up to the vessel bottom. This is called the critical speed for complete dispersion. Under these conditions the impeller action predominates over the action of gas sparging. That is, the average bubble size and the liquid flow pattern are controlled by the impeller action. A further increase in the impeller speed causes small bubbles to be recirculated back into the impeller region. The size of the recirculated bubbles increases with an increase in the impeller speed. Due to the recirculation of the bubbles the gas holdup steadily increases. This causes a reduction in the dispersion density in the impeller region, and the power number decreases. This is observed in the fourth region and is called the recirculation regime. The power consumption in the conventional mechanically agitated contactors is corelated empirically with the gas flow rate. Typical correlations (Tatterson, 1991) correlate the ratio of the power consumption in the gas state to the ungassed state (Pg/Po) with the superficial gas velocity (VG) or the gas flow number (VG/ND). It may be emphasized that, in the conventional mechanically agitated contactors, the gas flow rate and the impeller speed can be varied independently. In Contrast, in the gas-inducing-type mechanically agitated contactors the gas flow rate (gas induction rate) depends on the impeller speed. Therefore the power consumption behavior in these contactors is expected to be different. Saravanan et al. (1994) have discussed the mechanism of gas induction. When gas induction begins, three zones can be identified at the impeller plane (see Figure 1). These refer to the central zone of fluid recycle, middle zone of vortex where gas induction occurs at 0 e r e Y, and the outer zone outside the vortex which conveys gas (Y e r e R). This outer zone is submerged

+

+


The equation for torque becomes

τ ) τrg +

∫0YA

(

)

CDGWFL 2R2

(2πfsN)2r5 dr +

(

∫YRA

)

CDOWFL 2R2

(2πfsN)2r5 dr (12)

On integration it gives

τ ) τrg +

(

) (

2 C* DGWFLN

2R2

)

2 C* DOWFLN

[Y]6 +

2R2

[R6 - Y6] (13)

2/3 2/3 2 3 2 3 and C* where C* DG ) A CDGf s π DO ) A CDOf s π . Multiplying both sides by 2πN to get power terms and adjusting terms

P - (τrg)(2πN) 3

4

FLWN R

(RY)

) C* DG

6

) C* DO - (C* DO Figure 1. Schematic representation of vortex shape.

below the liquid surface. This portion of the impeller is exposed to the gas-liquid mixture as well as any cavities that may form behind the impeller blades. Extending the approach of the previous section, one can make the following additional assumptions: (1) For simplicity, the fluid recycled and the associated effects on the power consumption are assumed to be represented by a constant torque τrg acting at the eye of the impeller. (2) For the central zone of the impeller between 0 e r e Y, the associated drag coefficient is CDG. (3) The gas-liquid dispersion conveying zone is assumed to be between Y e r e R with an associated drag coefficient of CDO. (4) The impeller blades in the outer region, i.e. Y e r e R, are exposed to the gas-liquid dispersion. The effective density in the impeller region will therefore be reduced. This can also contribute to the reduction in the power number. The gas is induced in the middle portion and is carried away from the impeller. The effective density of the gas-liquid dispersion will be lowest in the middle zone, and it will increase as the radial location. In addition cavity formation may also contribute to the reduction in the dispersion density in the impeller region. Since the exact profile of gas holdup or the cavity size and shape is not known an empirical approach has to be adopted at this point. The effective density of the gas-liquid dispersion can therefore be taken as proportional to the liquid density and the square or radial distance along the impeller blade. Thus the density profile along the impeller is taken as

F ) AFL(r/R)2 for 0 e r e R At r ) R the above expression reduces to F ) AFL. The constant A in this case then represents the local value of the fractional liquid holdup (L).

[ (RY) ] Y - C* )( ) R 6

+ C* DO 1 -

6

DG

(14)

For simplicity let (C* DO - C* DG) ) C* DX. The value of (Y/ R)2 has been obtained by Saravanan et al. (1994).

() Y R

(

2

) (1 + kg) 1 -

)

2gS φV2

(15)

Let V2/2gS ) Fs

() () Y R

Y R

6

( (

) (1 + kg) 1 -

1 φFs

) (1 + kg)3 1 -

1 φFs

2

) )

(16)

3

Substitution into eq 14 and collecting terms appropriately

P - (τrg)(2πN) 3

4

FLWN R

[

3 ) C* DO - C* DX(1 + kg) 1 -

1 φFs

]

3

(17)

3 For convenience, let C*DX(1 + kg)3 ) C* DY(1 + kg) ) C*DY

P - (τrg)(2πN) 3

FLWN R

4

[

) C* DO - C* DY 1 -

1 3 φsF

]

(18)

4. Experimental Investigations were carried out in 0.57, 1.0, and 1.5 m i.d. gas-inducing-type mechanically agitated contactors. The schematic diagram of the contactor and the experimental setup are shown in Figure 2A,B. The design details of the impeller diameter and vessels are given in Tables 1A,B. Further, details pertaining to the equipment, instrumentation, and experimental procedure have been given by Saravanan et al. (1994). The impeller-stator pair has been specified in terms of the impeller diameter, as shown in Figure 2C. Impeller name abbreviations are listed in the Nomenclature section. The impeller Reynolds number varied from

+

+


Figure 2. (A) Schematic representation of experimental setup. (B) Various components of GIMAC: (S) submergence, (C1) impeller bottom clearance, (C3) interimeller clearance, (H) height of liquid level, (D) impeller diameter, (T) vessel diameter, (DS) stator diameter, and (HS) height of stator. (C) Schematic diagram of impeller and stator.

about 1 × 105 at the critical speed for gas induction to about 1 × 106 at the highest speed. 5. Results and Discussion Figure 4A shows a typical diagram of NP vs N (with S a parameter). When the impeller speed is increased from zero the value of NP increases with an increase in N and attains maximum and then decreases. The initial increase is probably due to the differential in the liquid level between inside and outside the standpipe. It is known that for conventional stirred vessels the value of NP is independent of submergence where the level differential does not occur. From Figure 4A it can be seen that the extent of maxima increases with an increase in the liquid submergence. For all values of

submergence, the maxima were found to indicate the critical impeller speed for gas induction. Therefore, a further increase in speed starts inducing gas by vortex formation in the impeller region. The depth of the vortex increases with the increasing impeller speed. Thus, as the speed increases, the portion of the impeller exposed to air increases. The power number therefore decreases. Simultaneously the rate of gas induction increases with an increase in the impeller speed, and hence the value of NP decreases. Cavity formation behind the impeller blades could also contribute to the reduction in the power number; however the presence of gas cavities behind the impeller blades could not be ascertained for the following reasons. (i) The gas-liquid dispersion issuing out of the

+

+


Figure 3. (A) Effect of impeller diameter on power consumption for a single impeller: T ) 1.5 m, S ) 0.3 m, C1 ) T/3. Symbol and impeller diameter (m): (+) 0.5; (*) 0.4; (0) 0.33; (×) 0.225. (B) Effect of impeller diameter on power number for a single impeller: T ) 1.0 m, S ) 0.3 m, C1 ) T/6. Symbol and impeller diameter (m): (+) 0.33; (*) 0.225; (0) 0.19. Table 1. Summary of Conditions for Runs during Gas Inductiona A: Single Impeller System mnemonic

C1 (m)

S (m)

N (rotations/s)

5719 5722 1019 1022 1033 1519 1522 1533 1540 1550

0.095/0.285 0.095/0.285 0.16/0.50 0.16/0.50 0.16/0.50 0.5/0.75 0.5/0.75 0.5/0.75 0.5/0.75 0.5/0.75

0.15/0.80 0.15/0.80 0.15/0.60 0.15/0.80 0.15/1.00 0.15/0.60 0.15/0.60 0.15/1.00 0.15/1.00 0.15/1.00

4.8/14.5 3.4/9.60 4.5/12.3 3.5/10.8 2.2/9.80 3.2/9.80 2.7/8.10 2.2/7.80 1.7/4.40 1.4/3.33

P (w) 20/760 20/1070 20/920 20/1120 50/1450 60/840 45/1350 60/1500 45/1450

B: Multiple Impeller System mnemonic

C1 (m)

C3 (m)

S (m)

N (rotations/s)

5719 5722 1019 1022 1033 1519 1522 1533 1540 1550

0.095/0.285 0.095/0.285 0.16/0.50 0.16/0.50 0.10/0.45 0.38/0.50 0.38/0.50 0.38/0.50 0.38/0.50 0.38/0.50

0.17/0.45 0.17/0.45 0.17/0.45 0.17/0.45 0.17/0.45 0.17/0.45 0.17/0.45 0.17/0.45 0.17/0.45 0.17/0.45

0.15/0.80 0.15/0.80 0.15/0.80 0.15/0.80 0.15/1.00 0.15/1.00 0.15/1.00 0.15/1.00 0.15/1.00 0.15/1.00

5.2/11.62 4.6/9.89 4.5/10.12 3.5/9.75 2.2/8.70 4.9/9.10 4.5/9.10 1.8/6.70 1.7/4.80 1.4/3.30

a

Table 2. Value of Various Constants for Single and Multiple Impeller Systems in the Absence of Gas Induction single PBTD PBTD-PBTD PBTD-PBTU PBTD-SBT PBTD-DT PBTD-PD PBTD-PU

80.33 147.27 153.90 307.98 384.44 93.77 86.66

1.12 2.44 2.23 4.67 5.84 1.41 1.26

trg

single PBTD PBTD-PBTD PBTD-PBTU PBTD-SBT PBTD-DT PBTD-PD PBTD-PU

1.767 0.762 0.390 0.789 0.927 0.485 0.15

a

impeller zone prevented visual observations, especially for multiple impeller systems where the second impeller helps in gas dispersion. (ii) The presence of the stator around the gas-inducing impeller prevented visual observations of the impeller blades during the process of gas induction. As the impeller speed increases the depth of the vortex formed increases. Due to the vortex the gasinducing impeller is partly exposed to air and partly to

NP

impeller combination

30/1130 47/1370 20/1020 20/1445 50/1650

Information given as min/max values.

CD

Table 3. Value of Various Constants for Single and Multiple Impeller Systems in the Presence of Gas Inductiona

P (w)

60/840 38/1350 60/1550 45/1730

impeller combination

φ

C*DO

C*DY

a

b

0.841 22.24 6.71 0.871 47.92 14.09 38.87 0.65 0.865 43.21 13.18 35.48 0.62 0.865 111.62 27.56 0.853 146.45 34.87 0.902 9.11 3.04 0.849 8.20 2.75 7.12 0.56

Upper impeller is PBTD in all cases.

the liquid. The part that is exposed to air increases as the impeller speed increases, and this contributes mainly to the reduction in power number. Thus the cavity formation mechanism is unlikely to occur especially for a single gas-inducing impeller. For multiple impellers, the cavity formation behind the bottom impeller cannot be ruled out. In the case of multiple impellers also, a reduction in the power number of the upper gas-inducing impeller contributes partly to an overall reduction in the power number of the system. A number of runs were performed for measuring the power consumption (prior to gas induction and after the gas induction) for various combinations of vessel diameter, impeller diameter, impeller bottom clearance,

+

+


Figure 4. (A) Effect of submergence on power number for a single impeller: T ) 1.5 m, D ) 0.5 m, C1 ) T/3. Symbol and submergence (m): (0) 0.15; (+) 0.4; (O) 0.5; (X) 0.6; (×) 0.7; (4) 0.8; (2) 0.9; (b) 1.0. (B) Effect of submergence on power number for a single impeller: T ) 1.5 m, D ) 0.5 m, C1 ) T/3. Symbol and submergence (m): (0) 0.15; (+) 0.4; (O) 0.5; (X) 0.6; (4) 0.7; (×) 0.8; (b) 0.9; (2) 1.0. (C) Effect of submergence on power number for a single impeller in 1.0 m vessel: T ) 1.0 m, D ) 0.33 m, C1 ) T/6. Symbol and submergence (m): (0) 0.15; (+) 0.3; (O) 0.4; (X) 0.5; (4) 0.6; (×) 0.7; (b) 0.8.

liquid height, impeller design, interimpeller clearance, and impeller speed. The range of variables covered for the single and multiple impeller systems is summarized in Table 1A,B. For brevity of report, the runs are identified with a four-digit number where the first two digits signify the

vessel size and the next two digits signify the impeller size. The run identification numbers also appear in Table 1. 5.1. Power Consumption Prior to Gas Induction. The power consumption of the impeller rotating in a liquid could be correlated by the following equation.

+

+

1590 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 3 4 P ) τr(2πN) + C* D(FWN R )

(11)

where

C*D ) CDf 2s π3

(7)

The regression analysis gave the following results:

τr ) 0

C* ) 92.56 for D g 0.33 m

τr ) 0

C* ) 84.65 for D e 0.225 m (21)

The CC was 0.962 for 110 DF, and the coefficients were highly significant at >99%. The same equation was also used for correlating the power consumption for multiple impeller system. Thus, regression was carried out using eq 11 for each impeller combination separately. The values of drag coefficient for different impeller combinations are given in Table 2. From this table it can be seen that the value of drag coefficient is more for complete radial flow impellers (DT and SBT) than complete axial flow impellers (PD and PU). Since the drag coefficient is related to the impeller power number, the increase in power number will increase the drag coefficient. The value of the impeller power number for different impeller designs is given by Ranade and Joshi (1989), Ranade et al. (1992), and Mishra and Joshi (1993). The value of the power number for different impeller designs is given in Table 2. From this table it can be seen that the value of the power number is in the following order: DT > SBT > PBTD > PBTU > PD > PU > single PBTD. 5.2. Power Consumption with Gas Induction. The most important variables influencing the power are the impeller diameter, impeller design, impeller geometry, and interimpeller clearance. 5.2.1. Single Impeller System. 5.2.1.1. Effect Impeller Diameter. Effect of impeller diameter on NP was studied using different PBTD impellers of diameters 0.19, 0.225, 0.33, 0.40, and 0.5 m in vessels of 0.57, 1.0, and 1.5 m. The results are shown in Figure 3A,B. The PBTD (T/3) has shown a minimum value (in 0.57, 1.0, and 1.5 m vessels) with respect to diameter. The increase in impeller diameter increases the impeller blade width which increases the radial component of the liquid velocity. So, the largest PBTD impellers (0.5 in 1.5 m, 0.33 in 1.0 m, and 0.225 in 0.57 m) have comparatively high power numbers. 5.2.1.2. Effect of Liquid Submergence. The effect of submergences on the power number was studied for the single impeller case. Figure 4A shows the effect of liquid submergence on the power number for a single PBTD impeller. From this figure, it can be seen that the power number increases with an increase in the liquid submergence. It was pointed out earlier that the maxima in the power number (Figure 4A) corresponds to the critical speed for the onset of gas induction. The critical speed for gas induction (NCG) was defined earlier by Saravanan et al. (1994). It is the impellerspeed at which the gas induction process just begins. In the present setup the critical impeller speed for gas induction was determined visually, as the speed at which bubbles were first induced in the liquid. Increasing the impeller submergence increases the hydrostatic head the impeller has to overcome before the gas induction can begin, and therefore the critical speed for gas induction increases with the increasing submergence.

Figure 5. Effect of impeller bottom clearance on power number for a single impeller in 1.0 m vessel: T ) 1.0 m, D ) 0.33 m, S ) 0.3 m. Symbol and impeller bottom clearance: (+) T/2; (*) T/3; (0) T/4; (×) T/6.

Since the critical impeller speed for gas induction increases with an increase in the liquid submergence, it was thought desirable to construct a plot of power number vs (N - NCG) which is shown in Figure 4B,C. It can be seen that there is practically no change in the value of power number with an increase in the liquid submergence. Similar observations were reported by Rewatkar and Joshi (1991) for the conventional mechanically agitated contactor. Thus the change in power number of the impeller with the impeller submergence could be solely attributed to the increase in the critical speed for gas induction. 5.2.1.3. Effect of Impeller Bottom Clearance. The effect of impeller bottom clearance was studied for the single impeller. Figure 5 shows the effect of impeller clearance on power number for a single PBTD. It can be seen that the power number increases with a decrease in the impeller clearance from the tank bottom. However, there is no significant change when the impeller clearance is less than or equal to T/3. The same observation was reported by Rewatkar and Joshi (1991) for conventional mechanically agitated contactors. The pitched blade turbines generate liquid flow in the axial (downward in the case of PBTD) as well as in the radial direction. When the impeller is away from the vessel bottom, the downward liquid flow continuously changes its direction so that the liquid flows along the wall. As the impeller clearance decreases, the change in direction becomes sharper and more energy is dissipated. For the smallest clearance the liquid flow almost hits the base so that practically all the kinetic energy is dissipated. As a result of energy dissipation during the change in direction an increase in NP was observed with a decrease in clearance. 5.2.1.4. Effect of Tank Diameter. The effect of tank diameter was studied in two ways. In the first

+

+


Figure 6. (A) Effect of vessel diameter (same D) on power number for a single impeller system: D ) 0.225 m, C1 ) T/3, S ) 0.3 m. Symbol and tank diameter (m): (0) 0.57; (+) 1.0; (*) 1.5. (B) Effect of vessel diameter (same D/T ratio) on power number for a single impeller system: C1 ) T/3, S ) 0.3 m. Symbol and tank diameter (m): (X) 0.57; (+) 1.0; (O) 1.5.

case, the same impeller (0.225 m) was used in all the vessels. In the second case, the same D/T (0.33) was maintained in all the vessels. Figure 6A,B shows the effect of power number on vessel diameter. From this figure it can be that there is no significant change in the power number by changing the vessel diameter which clearly shows the independence of the power number with respect to scale. 5.2.2. Multiple Impeller System. 5.2.2.1. Comparison of Impeller Designs. The effect of impeller design was studied for six different impeller designs, namely pitched blade downflow and upflow turbines (PBTD, PBTU), upflow and downflow propeller (PU, PD), straight blade turbine (SBT), and disk turbine (DT). In all cases the gas-inducing (upper) impeller was PBTD. Figure 7 shows the effect of impeller speed on power number for different impeller designs. It may be noted that the upper gas-inducing impeller is practically immersed in liquid, whereas the lower impeller rotates in the liquid phase. Therefore, the overall trend of power number with respect to impeller designs will be decided by the bottom impeller. Therefore, we see the following order with decreasing power number: PBTD-DT > PBTD-SBT > PBTD-PBTD > PBTDPBTU > PBTD-PD > PBTD-PU > single PBTD. The values of power number reported in Table 2 are expected to be useful in the design and scale up of such reactors. 5.2.2.2. Effect of Blade Angle. The blade angle was found to have a strong influence on power number (NP). The variation of NP, with blade angle as a parameter, as a function of the impeller speed for PBTD-PBTD and PBTD-PBTU impellers has been shown in Figures 8A and 9A. The impeller blade angle (horizontal to the axis) was varied from 30 to 90° for both PBTD and PBTU. It may be noted that the PBTD and PBTU having an angle of 90° is the straight bladed turbine. This turbine will not axially generate upward/ downward flow in the impeller stream. However, the

Figure 7. Effect of impeller designs on power consumption: T ) 1.0 m, D ) 0.225 m, C1 ) T/3, S ) 0.35 m. Symbol and impeller design: (]) single PBTD; (*) PBTD-PBTD; (+) PBTD-PBTU; (×) PBTD-PD; (4) PBTD-PU; (0) PBTD-SBT; (O) PBTD-DT.

notation PBTD(90) and PBTU(90) will be used for maintaining the continuity of the discussion on the effect of blade angle. For a given impeller of fixed diameter and blade width, the projected blade width (on vertical r-z plane) increases with an increase in the blade angle. Due to low projected blade width, less

+

+


Figure 8. (A) Effect of blade angle on power number for PBTD: T ) 1.0 m, D ) 0.225 m, C1 ) T/3, S ) 0.3 m. Symbol and blade angle (deg): (0) 30; (+) 45; (O) 60; (b) 90. (B) Effect of W/D ratio on power number for PBTD: T ) 1.0 m, D ) 0.225 m, C1 ) T/3, S ) 0.3 m. Symbol and W/D ratio: (0) 0.2; (+) 0.3; (O) 0.4. (C) Effect of number of blades on power number for PBTD: T ) 1.0 m, D ) 0.225 m, C1 ) T/3, S ) 0.3 m. Symbol and number of blades: (O) 4; (+) 6; (b) 8; (0) 12. (D) Effect of blade thickness on power number for PBTD: T ) 1.0 m, D ) 0.225 m, C1 ) T/3, S ) 0.3 m. Symbol and blade thickness (mm): (0) 2; (+) 3; (b) 5.

dissipation of energy occurs behind the impeller blades and therefore the PBTD-PBTU(30) has the lowest power number (NP). 5.2.2.3. Effect of Number of Blades. In order to study the effect of the number of blades, PBTD(45) and PBTU(45) impellers of 4, 6, 8, and 12 blades (W/D ) 0.3) were used in a 1.0 m i.d. vessel. The results are shown in Figures 8C and 9C. The power number increased with an increase in the number of blades. This

is obvious, because with an increasing number of blades, form dissipation increases and turbulence in the impeller region increases, also the interaction between the flow streams from different blades increases with an increase in the number of blades. Therefore, the impeller with 12 blades shows a higher power number than the impeller with four blades. 5.2.2.4. Effect of Blade Width. As pointed out by Uhl and Gray (1966), it is improper to use a blade width

+

+


Figure 9. (A) Effect of blade angle on power number for PBTU: T ) 1.0 m, D ) 0.225 m, C1 ) T/3, S ) 0.3 m. Symbol and blade angle (deg): (4) 30; (+) 45; (b) 60; (0) 90. (B) Effect of W/D ratio on power number for PBTU: T ) 1.0 m, D ) 0.225 m, C1 ) T/3, S ) 0.3 m. Symbol and W/D ratio: (0) 0.2; (+) 0.3; (b) 0.4. (C) Effect of number of blades on power number for PBTU: T ) 1.0 m, D ) 0.225 m, C1 ) T/3, S ) 0.3 m. Symbol and number of blades: (4) 4; (+) 6; (b) 8; (0) 12. (D) Effect of blade thickness on power number for PBTU: T ) 1.0 m, D ) 0.225 m, C1 ) T/3, S ) 0.3 m. Symbol and blade thickness (mm): (0) 2; (+) 3; (b) 5.

W in place of D in order to account for the effect of blade width on NP, because this implies that the power varies directly with blade width. In order to study this, the W/D ratio was varied from 0.2 to 0.4, keeping the impeller diameter (D) and the blade angle constant. The variation of NP with the W/D ratio for PBTD and PBTU is shown in Figures 8B and 9B. An increase in the blade width increases the form dissipation of energy. As already described previously, the radial component of liquid flow increases to a larger

extent when compared to the axial component with an increase blade width. At large blade width, the flow pattern will almost resemble that of a radial flow and the dissipation of energy behind the blades becomes very high. Due to this impeller with the larger blade width shows higher power number. 5.2.2.5. Blade Thickness. The effect of blade thickness is not very significant when compared to other parameters. Figures 8D and 9D show the effect of blade thickness for PBTD and PBTU. However, the slight

+

+


Figure 10. Effect of impeller diameter on power number for PBTD. (A) T ) 1.5 m, S ) 0.3 m, C1 ) T/3, C3 ) 0.17 m. Symbol and impeller diameter (m): (0) 0.5; (+) 0.4; (*) 0.33; (4) 0.225. (B) T ) 1.0 m, S ) 0.3 m, C1 ) T/3, C3 ) 0.17 m. Symbol and impeller diameter (m): (0) 0.33; (+) 0.225; (*) 0.19.

Figure 11. Effect of impeller diameter on power number for PBTU. (A) T ) 1.5 m, S ) 0.3 m, C1 ) T/3, C3 ) 0.17 m. Symbol and impeller diameter (m): (0) 0.5; (+) 0.4; (*) 0.33; (4) 0.225. (B) T ) 1.0 m, S ) 0.3 m, C1 ) T/3, C3 ) 0.17 m. Symbol and impeller diameter (m): (0) 0.33; (+) 0.225; (*) 0.19.

increase in NP with an increase in blade thickness is due to the edge effect of the impeller blades. That is, for a given impeller when the blade thickness is increased (keeping other dimensions constant), the lagging edge of the blade protrudes higher than the leading edge. This causes a two-fold effect. First, the projected area of the blade, on which the flow impinges directly, increases. Because of the protruding lagging edge, the flow has to take more turns (which are sharper

also). As a combined effect the power consumption increases. 5.2.2.6. Effect of Impeller Diameter. Effect of impeller diameter on NP was studied using different PBTD, PBTU, and PU impellers of diameters 0.19, 0.225, 0.33, 0.40, and 0.5 m in vessels of 0.57, 1.0, and 1.5 m. The results are shown in Figures 10-12. The PBTD, PBTU, and PU (T/3) have shown minimum values (in 1.0 and 1.5 m vessels) with respect to

+

+


Figure 12. Effect of impeller diameter on power number for PU. (A) T ) 1.5 m, S ) 0.3 m, C1 ) T/3, C3 ) 0.17 m. Symbol and impeller diameter (m): (0) 0.5; (+) 0.4; (*) 0.33; (4) 0.225. (B) T ) 1.0 m, S ) 0.3 m, C1 ) T/3, C3 ) 0.17 m. Symbol and impeller diameter (m): (0) 0.33; (+) 0.225; (*) 0.19.

Figure 13. Effect of submergence on power number for PBTD. (A) T ) 1.5 m, D ) 0.5 m, C1 ) T/3. Symbol and submergence (m): (0) 0.15; (+) 0.3; (O) 0.4; (×) 0.5; (X) 0.6; (×) 0.7; (O) 0.8; (b) 0.9; (4) 1.0. (B) T ) 1.0 m, D ) 0.33 m, C1 ) T/3. Symbol and submergence (m): (0) 0.15; (+) 0.3; (O) 0.4; (×) 0.5; (X) 0.6; (4) 0.7; (b) 0.8; (*) 0.9; (2) 1.0.

diameter. The increase in impeller diameter increases the impeller blade width which increases the radial component of the liquid velocity. So, the largest PBTD, PBTU, and PU impellers (0.5 in 1.5 m, 0.33 in 1.0 m, and 0.225 in 0.57 m) have a very high power number. 5.2.2.7. Effect of Submergence. The effect of

submergence on the power number was studied for PBTD, PBTU, and PU impellers. The critical impeller speed for gas induction increases with an increase in the liquid submergence. Therefore, the impeller power number increases with an increase in liquid submergence. However, from the plot of impeller power

+

+


Figure 14. Effect of submergence on power number for PBTU. (A) T ) 1.5 m, D ) 0.5 m, C1 ) T/3. Symbol and submergence (m): (0) 0.15; (+) 0.3; (O) 0.4; (×) 0.5; (X) 0.6; (4) 0.7; (b) 0.8; (*) 0.9; (]) 1.0. (B) T ) 1.0 m, D ) 0.33 m, C1 ) T/3. Symbol and submergence (m): (0) 0.15; (+) 0.3; (O) 0.4; (×) 0.5; (X) 0.6; (4) 0.7; (b) 0.8; (*) 0.9; (2) 1.0.

Figure 15. Effect of submergence on power number for PU: (A) T ) 1.5 m, D ) 0.5 m, C1 ) T/3. Symbol and submergence (m): (0) 0.15; (+) 0.3; (O) 0.4; (X) 0.5; (×) 0.6; (4) 0.7; (b) 0.8; (2) 0.9; (*) 1.0. (B) T ) 1.0 m, D ) 0.33 m, C1 ) T/3. Symbol and submergence (m): (0) 0.15; (+) 0.3; (O) 0.4; (X) 0.5; (×) 0.6; (4) 0.7; (b) 0.8; (2) 0.9; (*) 1.0.

number vs (N - NCG), it can be seen that there is practically no change in the value of power number with respect to submergence which is shown in Figures 1315 for PBTD, PBTU, and PU, respectively. 5.2.2.8. Effect of Impeller Bottom Clearance. The effect of impeller bottom clearance was studied for

PBTD, PBTU, and PU impellers. Figure 16 shows the effect of bottom clearance on power number for PBTDPBTU combination. It can be seen that the power number increases with a decrease in the impeller clearance from the tank bottom. However, there is no significant change when the impeller clearance is less

+

+


Figure 16. Effect of impeller bottom clearance on power number: T ) 1.0 m, D ) 0.33 m, C3 ) 0.17 m. Symbol and impeller bottom clearance: (0) T/2; (X) T/3; (4) T/4; (+) T/6.

than or equal to T/3. The same observation was reported by Rewatkar and Joshi (1991) for conventional mechanically agitated contactors. The pitched blade turbines generate liquid flow in the axial (downward in the case of PBTD) as well as in the radial direction.

When the impeller is away from the base, the downward liquid flow continuously changes its direction so that the liquid flows along the wall. As the impeller clearances decrease, the change in direction become sharper and more energy is dissipated. For the smallest clearance the liquid flow almost hits the base so that practically all the kinetic energy is dissipated. As a result of energy dissipation during the change in direction increases the NP was observed with a decrease in clearance. Recently Mishra et al. (1994) have studied this effect using LDA, and they found that impeller power number increases with a decrease in the impeller clearance from the bottom. 5.2.2.9. Effect of Interimpeller Clearance. The effect of interimpeller clearance was studied for PBTD, PBTU, and PU impellers. The clearance was varied in the range of 0.17-0.45 m. Figures 17-19 show the effect of interimpeller clearance for PBTD, PBTU, and PU impellers. From these figures it can be seen that the power number increases with an increase in the interimpeller clearance. This observation is in line with those reported for multiple impeller systems [Chiampo et al. (1991), Chiampo and Conti (1994)]. 5.2.2.10. Effect of Tank Diameter. The effect of tank diameter was studied in two ways. In the first case, the same impeller (0.225 m) was used in all the vessels. In the second case, the same D/T (0.33) was maintained in all the vessels. Figures 20 and 21 show the effect of vessel diameter on power number. It can be that there is no significant change in the power number by changing the vessel diameter which clearly shows that the impeller power number is independent of scale. 5.3. Data Correlation. 5.3.1. Single Impeller. The mathematical model for power consumption was described in section 3.3. The following equation was

Figure 17. Effect of interimpeller clearance on power number for PBTD. (A) T ) 1.5 m, D ) 0.5 m, S ) 0.3 m, C1 ) T/3. Symbol and interimpeller clearance (m): (O) 0.45; (4) 0.35; (+) 0.25; (X) 0.175. (B) T ) 1.0 m, D ) 0.5 m, S ) 0.3 m, C1 ) T/3. Symbol and interimpeller clearance (m): (O) 0.45; (4) 0.35; (+) 0.25; (X) 0.175.

+

+


Figure 18. Effect of interimpeller clearance on power number for PBTU. (A) T ) 1.5 m, D ) 0.5 m, S ) 0.3 m, C1 ) T/3. Symbol and interimpeller clearance (m): (O) 0.45; (4) 0.35; (+) 0.25; (O) 0.175. (B) T ) 1.0 m, D ) 0.33 m, S ) 0.3 m, C1 ) T/3. Symbol and interimpeller clearance (m): (O) 0.45; (+) 0.35; (4) 0.25; (X) 0.175.

Figure 19. Effect of interimpeller clearance on power number for PU. (A) T ) 1.5 m, D ) 0.5 m, S ) 0.3 m, C1 ) T/3. Symbol and interimpeller clearance (m): (O) 0.45; (+) 0.35; (4) 0.25; (X) 0.175. (B) T ) 1.0 m, D ) 0.33 m, S ) 0.3 m, C1 ) T/3. Symbol and interimpeller clearance (m): (O) 0.45; (+) 0.35; (4) 0.25; (X) 0.175.

derived

[

P - (τrg)(2πN) 3

4

FLWN R

]

[

) C* DO - C* DY 1 -

]

1 φsF

coefficients were obtained.

3

(22)

For a single impeller system the following regression

τrg ) 1.767 N m C*DO ) 22.24

φ ) 0.84 C* DY ) 6.71

The correlation coefficient was 0.953, with 456 DF.

+

+


Figure 20. (A) Effect of vessel diameter (same D) for PBTD: D ) 0.225; S ) 0.3, C1 ) T/3, C3 ) 0.17 m. Symbol and tank diameter (m): (0) 0.57; (+) 1.0; (*) 1.5. (B) Effect of vessel diameter (same D/T ratio): S ) 0.3 m, C1 ) T/3, C3 ) 0.17 m. Symbol and tank diamter (m): (0) 0.57; (+) 1.0.

Figure 21. (A) Effect of vessel diameter (same D) for PBTU: D ) 0.225, S ) 0.3, C1 ) T/3, C3 ) 0.17 m. Symbol and tank diameter (m): (0) 0.57; (+) 1.0; (*) 1.5. (B) Effect of vessel diameter (same D/T ratio): S ) 0.3 m, C1 ) T/3, C3 ) 0.17 m. Symbol and tank diameter (m): (0) 0.57; (+) 1.0; (*) 1.5.

Individual regression coefficients were also highly significant at >97%. Standard error of power estimate was 44 W, and the entire data was reproduced within +10% and -12% for all P > 40 W and C > 0.15.

The values of the drag coefficients can give us valuable information about the mechanism of the power consumption. In the presence of gas induction C* DO ) 2 /3(ACDOf 2s π3), and in the absence of gas induction C*D )

+

+


CDf 2s π3. The drag coefficients CD and CDO are characteristic of the impeller alone and are independent of the factors such as the cavity formation or the effective density of the gas-liquid mixture. Thus if they are equated, we get

4C*D f

2 3 sπ

3C* DO )

A)

2Af 2s π3

3C*DO 2CD* D

Substituting the values of C* DO and C* D from the regressed model, we get

A ) 0.36 Since A is representative of the fractional liquid holdup, this would imply that the value of gas holdup at r ) R would be around 0.64. Such a high value is unrealistic. This means that only the reduction in the effective density cannot account for all the reduction in the power number, and hence a part of the reduction in power number must be occurring due to the presence of cavities behind the impeller blades. 5.3.2. Multiple Impellers. For the case of multiple impellers, both impellers contribute toward the power consumption. By comparing the power numbers of multiple impeller system and single impeller system, we can calculate the relative contribution of both impellers. The contribution of the upper impeller is relatively small when the bottom impeller is a radial flow impeller such as DT or SBT. The contribution of the upper impeller can be significant when the lower impeller is an axial flow type like PBTD, PBTU, or PU. The contribution of the upper impeller is around 1040% of the total power consumption. The gas induced by the upper impeller is distributed by the lower impeller. The flow coming into and going out of the lower impeller region consists of a gas-liquid dispersion. In addition cavity formation may also be taking place behind the lower impeller blades. These two reasons are responsible for the reduction in the overall power number of the multiple impeller system. The regression of data was carried out for all the designs of lower impeller. The results are shown in Table 3. The individual coefficients were highly significant (>96%). The value of correlation coefficient was relatively low. This is because the interimpeller clearance is not included in the model. Therefore, the effect was included in the drag coefficient (C* DO) in the following form: b C* DO ) a(C3/D)

The values of a and b are given in Table 3. The values of the correlation coefficient were found to be 0.924, 0.936, and 0.908 for PBTD-PBTD, PBTD-PBTU, and PBTD-PU impellers. A sample parity plot for the correlation for the PBTD-PBTU system is given in Figure 22. It can be taken as a representative of a general multiple impeller system. When such a parity plot was made for all the data of the PBTD-PBTU system, it was observed that about 5% of the points showed significant deviation (up to 50%). Analysis of such deviating points did not reveal any systematic error. Hence these data points must indicate experimental error. The sources of such errors are under

Figure 22. Parity plot for PBTD-PBTU system.

investigation. When such data points were not used to evaluate the model parameters, the correlation coefficient of the regression model was high, and all the coefficients were highly significant. Comparing the values of C*DO for multiple impeller systems and C*D for single impeller system shows that the former values are lower when axial flow impellers are used. A similar argument can be made as that in section 5.3.1, and we can conclude that the power consumption is partly due to the reduction in the effective density and partly due to the presence of cavities behind the impeller blades. For the single impeller system the value of τrg was found to be significant. This is attributable to the turbulence generated at the eye of the impeller due to the gas-liquid oscillating interface. Before critical speed of gas induction, this interface and the associated turbulence are nonexistent. Hence, value of τr changes from zero to a finite value (∼1.8 N m) as the impeller state changes from non-gas-inducing to gas-inducing. The overall contribution of this factor to power consumption varies with the impeller speed. However, it must be recorded here that the maximum power consumption caused by this torque at the highest speed of impeller is ∼40 W. Therefore, resolution of τr would require more accurate measurements. From Table 3, it can be seen that the value of τrg gets reduced for all the multiple impellers. This could be due to the higher value of inertia of the system which dampens the unbalanced torque caused by the gas induction process. It can also seen from Table 3 that the value of drag coefficients C*DO and C* DY increases with an increase in the impeller power number. Thus, it shows a maximum value of drag for PBTD-DT combination. Since in GIMAC the drag coefficients are related to the impeller power number the above results are obvious. From the forgoing discussion it can be seen that the mathematical model satisfactorily explains all the experimental data reported in the figures. In the

+

+


present work, all the experiments were carried out on an air-water system. Therefore it is not possible to comment on the parameters (such as viscosity and density) which affect the values of the constants (τrg, C*DO, C* DY). The variation of these constants with respect to the impeller design is the subject of investigation here. From Tables 2 and 3 it can be seen that the performance of the various impellers can be completely characterized by these constants. 6.0. Conclusions (1) Mathematical models have been developed in the absence and presence of gas induction. The power reduction in the presence of gas has been modeled through the vortex formation. An increase in the impeller speed increasingly exposes the impeller and hence reduces the power number. (2) Power consumption has been measured for single and multiple impellers in 0.57, 1.0, and 1.5 m i.d. vessels over a wide range of impeller design, diameter, submergence, and clearance. The mathematical model satisfactorily explains all the experimental measurements.

ω ) angular velocity of impeller (rad s-1) τ ) torque (N m) θ ) angular coordinate (-) Subscripts a ) average at ) atmospheric b ) blade c ) critical state for gas induction f ) fluid, friction g ) gas, gassed state i ) impeller l ) liquid m ) conformity max ) maximum o ) outer edge of impeller r ) recycle to impeller s ) slip sp ) standpipe x ) vortex related v ) vessel Superscripts

Nomenclature a ) constant head reduction at impeller edge C1 ) impeller clearance from tank bottom (m) C3 ) interimpeller clearance (m) CC ) correlation coefficient C ) drag coefficient of impeller CH ) head coefficient D ) impeller diameter (m) DS ) diameter of stator (m) DT ) disk turbine DF ) degree of freedom F ) drag force on impeller (N) Fs ) froude number based on submergence ) V2/2gS (-) f ) correction factor g ) gravitational acceleration (m s-2) GIMAC ) gas-inducing mechanically agitated contactor H ) liquid height (m) k ) head gain coefficient at impeller periphery (-) N ) impeller rotational speed (Hz) N ) impeller rotational speed (rotations/s) Np ) power number ) P/FN3D5 (-) p ) static pressure (Pa) PBTD ) pitched turbine downflow PBTU ) pitched turbine upflow PU ) propeller upflow PD ) propeller downflow R ) impeller radius (m) S ) impeller submergence in ungassed liquid (m) SBT ) straight blade turbine T ) vessel diameter (m) V ) impeller tip velocity (m s-1) WA ) actual width of impeller (m) WP ) vertical projected width of impeller (m) Y ) radius of intersection of paraboloid of vortex at impeller plane for power consideration (m) Greek Letters R ) aspect ratio of impeller ) W/D (-) ) fractional holdup φ ) vortexing constants (-) F ) density (kg m-3) µ ) viscosity (Pa s)

* ) scaled parameter

Literature Cited Arbiter, N.; Harris, C. C.; Yap, R. J. Hydrodynamics of Flotation Cells, Trans. Soc. Min. Eng. AIME 1969, 244, 134-148. Chimpo, F.; Guglielmetti, R.; Manna, L.; Conti, R. Gas-Liquid Mixing in a Multiple Impellers Stirred Vessel. Proceedings of the 7th European Conference on Mixing, KIAV: Brugge, Belgium, September 18-20, 1991; p 333-340. Hixon, A. W.; Baum, S. J. Agitation-Power Requirements of Turbine Agitators. Ind. Eng. Chem. 1942, 34 (2), 194-208. Hudcova, V.; Machon, V.; Nienow, A. W. Gas-Liquid Dispersion with Dual Rushton Turbine Impellers. Biotech. Bioeng. 1989, 34, 617-628. Joshi, J. B. Modifications in the design of Gas inducing Impellers. Chem. Eng. Commun. 1980, 5, 109-114. Mishra, V. P.; Joshi, J. B. Flow Generated by a Disc Turbine: part III: Effect of Impeller diameter, Impeller location and comparison with other Radial and Flow Turbines. Chem. Eng. Res. Des. 1993, 71, 563-570. Mishra, V. P.; Kumar, P.; Joshi, J. B. LDA Measurements of Flow Generated by Multiple Pitched Blade Turbine Impellers. Proceedings of the 8th European Conference on Mixing; BHRA: Cambridge, MA, 1994; p 465-468. Mundale, V. D.; Joshi, J. B. Optimization of Impeller Design for Gas Inducing Type of Agitated Contactor. Can. J. Chem. Eng. 1995, 73, 6-17. Raidoo, A. D.; Raghava Rao, K. S. M. S.; Sawant, S. B.; Joshi, J. B. Improvements in Gas Inducing Impeller Design. Chem. Eng. Commun. 1987, 54, 241-264. Ranade, V. V.; Joshi, J. B. Flow Generated by Pitched Blade Turbine I: Experimental. Chem. Eng. Commun. 1989, 81, 197224. Ranade, V. V.; Mishra, V. P.; Saraph, V. S.; Deshpande, G. B.; Joshi, J. B. Comparison of Axial Flow Impellers Using Laser Doppler Anemometer. Ind. Eng. Chem. Res. 1992, 31, 237123379. Rewatkar, V. B.; Joshi, J. B. Role of Sparger Design in Mechanically Agitated Gas-Liquid Contactor - Part I: Power Consumption. Chem. Eng. Technol. 1991, 14, 333-347. Saravanan, K.; Joshi, J. B. Gas Inducing Type Mechanically Agitated Contactors: Hydrodynamic Characteristics of Dual Impeller System. Ind. Eng. Chem. Res. 1995, 34, 2499-2514. Saravanan, K.; Mundale, V. D.; Joshi, J. B. Gas Inducing Type Mechanically Agitated Contactors. Ind. Eng. Chem. Res. 1994, 33, 2221-2240.

+

+

1602 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 Tatterson, G. B. Gas Dispersion in Agitated Tanks. Fluid Mixing and Gas Dispersion in Agitated Tanks; McGraw Hill, Inc.: New York, 1991; Chapter 6, pp 450-459. Uhl, W. V.; Gray, J. B. Mixing, Theory and Practice; Academic Press: New York, 1966; Vol. I, p 1. Zundelevich, V. Power Consumption and Gas Capacity of Self Inducing Turbo Aerations. AIChE J. 1979, 25, 763-773.

Received for review January 23, 1996 Accepted February 5, 1996X IE9503353 X Abstract published in Advance ACS Abstracts, April 1, 1996.

Power Consumption in Gas-Inducing-Type Mechanically Agitated

Recommend Documents