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Analysis of the Fluid Dynamic Behavior of the Liquid and Gas Phases in Reactors Stirred with Multiple Hydrofoil Impellers. D. Pinelli, and F. Magelli*...
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Ind. Eng. Chem. Res. 2000, 39, 3202-3211

Analysis of the Fluid Dynamic Behavior of the Liquid and Gas Phases in Reactors Stirred with Multiple Hydrofoil Impellers Davide Pinelli and Franco Magelli* Department of Chemical, Mining and Environmental Engineering, University of Bologna, viale Risorgimento 2, 40136 Bologna, Italy

Liquid- and gas-phase macromixing behavior was studied in gas-liquid high-aspect-ratio reactors stirred with multiple hydrofoil impellers pumping downward. Water, a sodium sulfate solution, and poly(vinylpyrrolidone) solutions of viscosity up to 110 mPa‚s were used as the liquid. For characterizing the liquid phase, mixing time experiments were conducted at various operating conditions, while detecting the response curves at several positions inside the tank. Comparison of the experimental curves with the theoretical ones provided by simple fluid dynamic models showed that the axial dispersion model is quite acceptable. The influence of impeller speed, gas flow rate, and viscosity on the model parameter was studied, and dimensionless relationships are given. The gas behavior was studied by means of the RTD and modeled with the axial dispersion model, which proved good for water and acceptable with coalescence-inhibiting electrolyte solutions. The model parameter dependence on the operating conditions was studied. Comparison between hydrofoil impellers and radial Rushton turbines is also attempted. 1. Introduction Gas-liquid stirred reactors are widely used in the chemical process industry as well as in biotechnology and have been the subject of many investigations.1 Great attention has been paid in the past decade to the development and the characterization of new impellers that would enhance and optimize gas-liquid contact in the tanks with respect to the traditional Rushton turbines and make this operation more flexible in a wider range of conditions.2,3 Multiple impellers are often used in industrial applications because they provide better gas utilization and higher heat-transfer surface per unit volume in comparison with the single-impeller geometry. The multiple-impeller arrangement, which was originally adopted for the Rushton turbines, has been extended to the novel impellerssused either in sets of identical elements or in mixed configurations of novel and traditional impellers, which gives rise to a variety of possible geometries. Whereas much information exists in the literature for the features of multiple Rushton turbines, it is scant for the configurations based on novel impellers. The available studies on the latter regard power consumption and holdup,4-9 mixing time,4,7,8,10,11 and mass-transfer coefficients.6,8,12 Suitable information about the two-phase fluid dynamics and the modeling of their behavior is necessary for design purposes and for predicting equipment performance. In this area, even less data are reported, namely, the description of gas and liquid flows in axial impellers configuration,11,13,14 the (unsatisfactory) performance of a compartment model in describing liquid flow behavior,15 and the estimate of (the order of magnitude of) the axial dispersion coefficient of the gas phase.13 The insufficiency of such information prompted * Corresponding author. Fax: +39-051-581200. Telephone: +39-051-2093147. E-mail: [email protected].

an investigation on these aspects that was focused on multiple, identical hydrofoil impellers. In this paper, the results of a study on liquid- and gas-phase behavior and their description by means of fluid dynamic models are presented. The hydrodynamics of these systems is rather complex because of phase interactions: gas bubbles are entrained by the liquid to an extent that depends on bubble size, liquid properties, flow pattern, impeller geometry, etc., while the bubble swarm, in turn, can interfere with the flow patternsthus either increasing or decreasing liquid mixing time according to the operating conditions. To simplify the study of these systems, the analysis of each of the two phases has been conducted separately, while considering the influence of the other as an external, independent factor. For both phases, modeling has been attempted by resorting to simple models with the aim of discussing the influence of the main physical and operating parameters and providing simple means for preliminary quantitative equipment evaluation. 2. Experimental Section 2.1. Equipment and Experimental Conditions. The experiments were conducted in a cylindrical tank (T ) 48 cm diameter, V ) 259 L working volume) having aspect ratio H/T ) 3. The tank had a flat bottom and four vertical standard baffles. Agitation was provided with multiple identical, evenly spaced hydrofoil impellers (diameter D/T ) 0.40) mounted on the same shaft (the high-solidity-ratio Lightnin A315 impellers were selected). The experiments were carried out with three impellers (H/T ) 3 tank configuration) and, for selected conditions, with two impellers (H/T ) 2). The lowest impeller was placed at T/2 above the tank baseswhose value is slightly higher than the reported optimum one.16 The main features of the equipment are shown in Figure 1. The experiments were performed in semibatch conditions at room temperature and atmospheric pressure.

10.1021/ie000216+ CCC: $19.00 © 2000 American Chemical Society Published on Web 07/29/2000

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cL(t) - cL(0) cL(∞) - cL(0)

Figure 1. Experimental tank and setup. A-F: liquid measurement/sampling elevations.

The liquid batches were demineralized water (coalescent system) and aqueous poly(vinylpyrrolidone) solutions (PVP; Newtonian behavior with viscosities 1, 10, 27, and 110 mPa‚s) for the investigation on the liquid phase and demineralized water, a 0.2 M sodium sulfate solution (noncoalescent behavior), and a PVP solution (viscosity 5 mPa‚s) for that on the gas phase. Air was fed to the system through a ring sparger (DS/D ) 0.7) located below the bottom impeller. The working conditions were N ) 4-9.7 s-1 and QG ) (0.7-5.0) × 10-3 m3/s (which is equivalent to UG ) 0.0039-0.028 m/s or 0.1-0.9 vvm, based on the H/T ) 3 configuration). Surface aeration was negligible in these conditions. The Froude number was always greater than 0.1. In addition to the experiments aimed at characterizing the liquid- and gas-phase mixing behavior, which will be discussed in the following sections, the overall power consumption and gas holdup were evaluated with standard techniques.5,17 2.2. Mixing Time Measurements. The mixing behavior of the liquid phase was investigated by means of the dynamic technique, which is usually adopted for determining the mixing timeseither in the absence or in the presence of the gas. It was based on a rapid injection of an electrolytic tracer solution (KCl in this study) at the top of the vessel about 5 cm below the liquid surface and on the measurement of the resulting concentration at several vertical positions (shown in Figure 1) as a function of time. In the case of no aeration, a standard two-electrode conductivity probe was simply inserted in the vessel for this purpose. Because gas bubbles disturb the liquid conductivity measurement on aeration, a small portion of liquid was withdrawn continuously from the measurement point at constant flow rate after separating the gas and was subjected to measurement in a continuous mode. Both techniques gave equal results under conditions of no aeration or limited bubble influence. The experimental curves obtained at several (vertical and radial) positions were used for determining the time to get 95% homogeneity (defined as mixing time t95), i.e., the time needed for the ratio

to become equal to 1 ( 0.05. They were also used for matching the theoretical curves provided by simple flow models and, thus, for assessing the model parameter by means of a best-fit procedure. Several curves were always determined for each operating condition in order to evaluate the reproducibility of the response curves, the mixing time, and the flow model parameters. The average values of the last two parameters were retained for subsequent elaboration. 2.3. Gas-Phase Characterization. The gas-phase behavior was characterized by means of RTD measurements. The details of the technique are the same as those given in previous papers18,19 and are only summarized here. A small amount of a gas tracer (methane) was rapidly injected into the air stream at the sparger inlet. Part of the gas was sampled at the top of the vessel through an inverted funnel (3.6 cm in diameter) located at the liquid surface and analyzed continuously in a flame-ionization detector. Therefore, the actual response of each experiment (apparent RTD) includes the contribution of the stirred gas-liquid system proper and that given by the funnel and the line from the funnel to the detector. The RTD of the latter devices was evaluated by means of single tracer pulses into the funnel and convoluted with the model equations (see section 3.2). Besides, the experimental response curve is affected by tracer mass transfer between the gas and the liquidsessentially in the part after the peak.18,20 Although tracers less soluble than methane would have reduced this effect,21,22 the simple procedure available to account for this artifact18,19 was used throughout the whole investigation (see section 3.2). To increase reliability, many curves were always determined and averaged for each experimental condition: 8-15 for the apparent tank RTD and 3-5 for the (much more reproducible) funnel-and-line RTD. Usually, the whole set of measurements was repeated two or three times for each condition for better evaluation (for the most critical conditions such as those in the H/T ) 2 configuration, they were done in quadruplicate). 3. Fluid Dynamic Models and Data Treatment 3.1. Flow Models for the Liquid (Batch Conditions). Two well-known flow models were used for this analysis of the liquid-phase behaviorsin a way similar to that used with radial turbines.23 They are the axial dispersion model and the cascade of ideal stages with backmixing; stages of equal volume were considered for the latter. For a batch system, the following mass balance equations and boundary conditions describe the dimensionless response to a pulse of n0 moles of a passive tracer into the one end of the system:24

Cascade of Ideal Stages with Backmixing (CISB) jth stage:

V dcj ) fcj-1 + fcj+1 - 2fcj J dt

(1)

1st stage:

V dc1 ) n0δ(t) + fc2 - fc1 J dt

(2)

last stage:

V dcJ ) fcJ-1 - fcJ J dt

(3)

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Axial Dispersion Model (ADM) ∂cL ∂2cL ) DeL 2 ∂t ∂z

(4)

initial and boundary conditions: cL(0,z) ) 0 cL|z)0 - DeL

|

∂cL ) n0δ(t) ∂z z)0

(5)

|

∂cL )0 ∂z z)H (6)

and

The dimensionless solutions of the above equations subject to the given initial and boundary conditionss namely, Cj(θ) and CL(θ,ζ) for the two models, respectivelys can be found in the literature; see Sawinski and Hunek25 for the CISB and Siemes and Weiss26 for the ADM. 3.2. Flow Model for the Gas Phase (ContinuousFlow Conditions). The one-dimensional axial dispersion model was adopted to describe the behavior of the gas phase. When tracer mass transfer between the gas and the liquid can be neglected,18,19 the following dimensionless mass balance equation holds good for a gaseous passive tracer:

∂2CG ∂CG ∂CG 1 ) ∂θ PeG ∂ζ ∂ζ2

(7)

For an ideal pulse disturbance at the stream inlet and nil tracer turbulent transport at the exit, which implies negligible surface aeration,17 the following initial and boundary conditions apply:

CG(0,ζ) ) 0 CG|ζ)0 -

|

1 ∂CG ) δ(θ) PeG ∂ζ ζ)0

and

(8)

|

∂CG ) 0 (9) ∂ζ ζ)1

The solution of eq 7 subject to the conditions (8) and (9) is available in the literature.27 The model parameter PeG was thus determined by comparing the experimental data with the theoretical curves by means of a best-fit technique. For dealing with the contribution given to the experimental curve by the funnel and the transfer line, the following strategy was adopted:18,19,28,29 each tentative model curve was convoluted with the funnel-and-line RTD, thus obtaining an "implemented” model curve; this last was then compared with the experimental one; the PeG value providing the least departure between the curves was retained as the most suitable one. To circumvent the need for accounting for tracer solubility in the liquid, the quantitative comparison between the experimental and theoretical curves was limited to the ascending part of the normalized curves.18 4. Results and Discussion 4.1. Qualitative Observations. Some preliminary experiments were conducted with water to optimize the tracer injection device and the injection point as well as to obtain qualitative information about the mixing liquid behavior of the system. For this purpose a few cubic centimeters of a colored tracer were injected into the reactor at its top. The spread of the tracer in the liquid was recorded by means of a VCR and viewed at

a reduced speed. As a result, a multiple-point injection device was adoptedswith three short needles discharging horizontally. It was also found that the tracer had to be injected at about 5 cm below the liquid surface to prevent excessive stagnation around the injection point and ensure proper liquid tracing. This drawback is the result of the limited liquid turbulence existing close to the surface in the case of axial impellers, as discussed by Baudou et al.14 Interestingly, liquid compartmentalization was noticed on aeration in some conditions that was particularly evident between the middle and the bottom impellers. Such zoning was hardly noticed in the absence of aeration. At the lowest impeller speed, the bubble size was rather heterogeneous12 regardless of the liquid used. At higher rotational speed, the average bubble size was on the order of a few millimeters in water; the bubble-size distribution became essentially bimodal with the PVP solutions,4 with the smaller fraction having the size of a fraction of a millimeter. Interestingly, big air bubbles were intermittently formed at high gas flow rate with the PVP solutions and detached and rose rapidly around the shaft, thus bypassing the impellers and avoiding their dispersing action. When the liquid level was adjusted to H/T ) 1, 2, or 3, the behavior of one, two, or three impellers could be observed separately at least with water and the variety of regimes originally identified by McFarlane et al.30 were recognized in these cases too: they range from flooding to complete dispersion passing through the asymmetric, radial, and competitive regimes and also include the precession of the asymmetric one. The flow patterns observed with the single impeller (H/T ) 1) fully confirm previous findings.30 With either the H/T ) 2 or 3 configurations, some differences between the bottom impeller and the upper one(s) were apparent (in the case of H/T ) 3, the middle and the uppermost impellers behave very similarly): despite the high impeller spacing, the bottom impeller is somehow affected by the presence of the top ones, with the most meaningful effect being the narrowing of the competitive regime in favor of the asymmetric and radial ones. 4.2. Liquid Mixing Times at Low Viscosity. The response curves to tracer injection were rather different, depending on the elevations at which they were detected. They always exhibited a marked overshoot for measuring positions nearby the uppermost impeller (Figure 2a), a gradual increase in the lower third of the reactor (Figure 2b), and intermediate behavior in the middle (in the figures the concentration was normalized by the difference between the concentration values at t ) 0 and at the asymptote). Reproducibility of the curves was not always complete under gassing conditions, possibly because of the inherent flow dynamic instability of impellers pumping downward.3,16 Despite these two facts, the values of the mixing time t95 were fairly similar regardless of the detection point (Figure 3). This behavior is fairly different from the results of Cronin et al.31 for dual Rushton turbines where marked compartmentalization occurs and is probably typical of multiple axial flow impellers at large. Moreover, there were unappreciable differences in the curve shape and mixing time for the measurements made at different radial positions. In the case of no aeration, the mixing time with the A315 configuration is about half that measured with

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Figure 4. Backflow rate (CISB model) determined from the experimental curves as a function of elevation for various impeller speeds; water, vvm ) 0.

Figure 2. Dimensionless concentration as a function of time. QG ) 0; N ) 5 s-1; water: O, experimental curve; - - -, theoretical curve of the axial dispersion model (ADM). (a) ζ ) 0.33; (b) ζ ) 1.

Figure 3. Mixing time t95 as a function of elevation for N ) 5 s-1 and N ) 7 s-1; water, vvm ) 0.

three Rushton turbines at equal specific power consumption (in the range Pg/V ) 0.15-1.0 kW/m3). Generally, the aeration rate increases t95. While this effect is most pronounced at high impeller speeds, N and QG affect the mixing time less clearly at lower speeds. A more complete discussion of these aspects will be developed in section 4.4 in terms of the model parameter. 4.3. Model Discrimination for the Liquid Phase. As anticipated above, the suitability of two simple flow models (namely, the CISB and the ADM) was investigated for describing the liquid behavior in tanks stirred with multiple axial impellers. This analysis was performed for selected operating conditions and vertical positions. Both j and ζ were computed starting from the liquid surface at the top of the tank. Under the hypothesis of limited impeller interaction, which is a consequence of the high impeller spacing, the number of ideal stages in the CISB model was taken to

be equal to the number of impellers (i.e., J ) 3). To justify the CISB model, equal response curves would have been expected for each impeller zone. Essentially the same value of the model parameter f should have resulted regardless of the measuring elevation ζ. Neither of these conditions was fulfilled. An example of f values obtained at various rotational speeds as a function of ζ is shown in Figure 4 for the case QG ) 0. Differences in f as high as 50% are apparent. Essentially the same differences were observed on aeration, which confirms previous findings of Otomo et al.15 It is also worth mentioning that the model is inherently incapable of providing the overshoot in the response curve at the uppermost measuring positions (Figure 2a), and even at high ζ values (i.e., low elevations), the theoretical curves do not match the experimental ones very well. On the contrary, the theoretical curves of the axial dispersion model match the experimental ones at all measuring positions very satisfactorily: examples for ζ ) 0.33 and 1.0 are given in parts a and b of Figure 2. In addition, the best-fit values of the model parameter, DeL, are essentially independent of the axial coordinate under either no aeration (Figure 5a, with a maximum error of 10%) or aeration (Figure 5b, maximum deviation of 17% at N ) 7 s-1). The axial dispersion coefficient is also independent of the radial coordinate (Figure 6). Therefore, the ADM model proves to be adequate for preliminary modeling purposes. The influence of the working conditions on the model parameter is examined below. Two further aspects are worth mentioning about the ADM. The overshoot exhibited at low ζ values is always (slightly) higher for the experimental curve than for the theoretical one: this seems to depend on the way the tracer is introduced into the tank (“point” injection) as well as on the noninfinite value of the radial dispersion mechanisms and coefficient. Besides, dimensionless measurement positions ζ approximately equal to 0.4 are to be avoided: indeed, a very flat maximum is exhibited by the curves in these cases that cannot be easily recognized in the experimental ones because of data noise, which makes parameter evaluation very inaccurate. For this reason, measurements were not performed at this elevation. 4.4. Influence of the Working Conditions and Viscosity on the Axial Dispersion Coefficient. The influence of rotational speed and aeration rate on the parameter of the ADM can be conveniently examined by keeping one parameter constant and varying the other. Without aeration and with water, the axial

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a

a

b

b

Figure 5. DeL as a function of rotational speed and elevation. Water; (a) QG ) 0 L/min; (b) QG ) 80 L/min.

Figure 6. Influence of the radial position on DeL for various impeller speeds. Water, ζ ) 0.75.

dispersion coefficient is proportional to the impeller speed, with this finding confirming previous results for the turbulent regime.32 Therefore, the dimensionless parameter DeL/ND2 seems to be appropriate for representing the data. This dependence remains identical at low gassing rates, while diminishing at an increase in QG; for gassing rates greater than 0.5 vvm, the axial dispersion coefficient becomes almost independent of rotational speed (Figures 5b and 7a). Overall, the aeration rate reduces the axial dispersion coefficient. As is apparent from Figures 6 and 7b, its extent decreases with an increase in QG at high agitator speeds, whereas a partial reversal of this effect takes place at low N. These facts can be explained with the hampering effect of the gas bubbles on liquid mixing at high rotational speed and the competition between stirring and aeration in determining the state of mixing at lower speeds.33 Despite this, the full symptoms of flooding were not recognizedsnot even at the highest QG and the lowest N values investigated in this paper.

Figure 7. Combined influence of the rotational speed and gas rate on the axial dispersion coefficient: (a) absolute values; (b) relative values.

Figure 8. Dimensionless axial dispersion coefficient for the liquid as a function of Reynolds number. Line: interpolation of the data for ungassed conditions. Data points: gassed conditions.

As regards the influence of viscosity on the liquid axial dispersion coefficient, a simple way for discussion is to plot DeL/ND2 as a function of the Reynolds number. For unaerated conditions, this dimensionless parameter increases with Re up to Re ) 4 × 104, and then it becomes constant. Both this behavior and the data are in agreement with what had been determined previously.32 Only the interpolating line of the data is shown in Figure 8, while the data themselves are omitted for clarity. Instead, the behavior for gassed conditions is more complex. The data show as almost independent sets in Figure 8. In fact, at the lowest viscosity (water and dilute PVP solutions) the aeration rate affects axial dispersion slightly and DeL is always lower than the values obtained at QG ) 0 (see section 4.4). On the contrary, with the most viscous solutions the influence of the gas rate on the dispersion coefficient becomes

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Figure 9. Fit of eq 9.

more pronounced and, more important, DeL values are higher than those obtained in the absence of gas. Overall, DeL dependence on viscosity is fairly limited, and a single value DeL/ND2 ) 0.11 can be retained for quick estimates for this reactor configuration ((30% maximum error), regardless of the values of N, QG, and viscosity. A peculiar relationship does exist between the dimensionless axial dispersion coefficient and holdup: at any viscosity, the former increases with the latter up to a maximum, and then it decreases. The extent of this overall change is higher with higher viscosity (up to the maximum value investigated of 120 mPa‚s). This is likely related to the variable structure of the holdup with operating parameters and viscosity5,34 and its complex relationship with the overall circulation, coalescence rate, and liquid entraining capability for the gas. Because of this complexity, no simple mathematical link between these parameters was found. If the mentioned nonmonotonic relationship is ignored, the following correlation can be determined:

DeL/ND2 ) 17 exp(-10.5)

(10)

This equation, whose fit with the experimental data is shown in Figure 9, can be used for liquid-phase axial dispersion estimates (average error (17%). 4.5. Power Draw and Gas Holdup. Overall, the relative power draw Pg/Pu of the multiple-impeller arrangement is equal to that reported for a single impeller. However, the slightly different behavior detected between a single and two or three impellers suggests that there exists a limited degree of asymmetry in the multiple-impeller operation even with hydrofoils. PVP solutions of low viscosity exhibit holdup values higher than those of water at equal working conditions, which is reminiscent of that of noncoalescent systems, while at higher viscosity, the holdup becomes lower. This behavior is similar to that already noticed with Newtonian35 and pseudoplastic liquids.36 Increasing the viscosity is apparently responsible for the change in bubble size and bubble-size distribution,34 which in turn affects holdup. Overall, holdup values are in good agreement with those found previously.5 The following relationship interpreted all of the data satisfactorily:

 ) A1(Pg/V)0.333(UG)0.667(η/ηref)-δ

(11)

with A1 ) 0.149 and δ ) 0 for demineralized water and A1 ) 0.566 and δ ) 0.433 for the PVP solutions. The given set of parameters for water are practically identi-

Figure 10. Examples of experimental response curves for the H/T ) 2 configuration; water; QG ) 215 L/min, N ) 9.6 s-1.

cal with those found by van’t Riet37 and very similar to those reported by Pinelli et al.5 and Bouaifi et al.7 The value of the exponent δ for viscous liquids was reported to be 0.13 for CMC solutions34 and 1.17 for glycerol:35 no explanation for these differences is possible at present, apart from recognizing that they reflect differences in the superficial properties of the systems. Similarly, the equation proposed by Smith38

 ) A2(Re × Fr × Fl)a(D/T)1.25

(12)

does fit the data with A2 ) 3.7 × 10-3 for water and A2 ) 1.3 × 10-2 for all of the PVP solutions, while the exponent a is equal to 0.45 regardless of the liquid. 4.6. Preliminary Gas RTD Measurements. Preliminary RTD measurements were carried out with demineralized water to define some details of the experimental technique, while qualitatively checking the validity of the model. The reproducibility of the single experimental curves determined for each set of measurements is worse with the hydrofoil impellers (Figure 10) than with the Rushton turbines19sslightly better than shown in Figure 10 at high impeller speed (especially with H/T ) 3) and worse at low N and QG with H/T ) 2. This is likely due to the above-mentioned flow instability typical of axial impellers pumping downward. Therefore, an average curve was calculated for each set of experimental response curves. It was found that the model parameter PeG obtained from this average curve was approximately equal to the mean of the PeG values determined from the single curves (average and maximum deviation 13% and 23%, respectively). This curve-averaging technique was, therefore, always adopted for PeG determination. The exit gas concentration and the resulting PeG values were slightly different when the traced gas was withdrawn near the shaft, at the vessel wall, or midway between these extreme positions because of the uneven local bubble velocities.13 Because the average of the PeG values resulting from the curves measured at the two extreme positions along the radius was generally equal to the value determined with the probe placed midway,19 all of the routine measurements were performed at that intermediate position. Overall, the fit between the “implemented” theoretical curve (that is, the model curve convoluted with the withdrawal system response) and the “average experimental” one is fairly good, especially for the experiments with water. At this stage, therefore, the axial dispersion model seems to be acceptable for describing the timeaverage behavior of the gas in this equipment. More

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Figure 11. PeG values from RTD measurements as a function of πND and QG. Demineralized water and H/T ) 3 configuration. - - -, average, interpolating line for three Rushton turbines.19

Figure 12. PeG values from RTD measurements as a function of πND and QG: 0.2 M sodium sulfate solution and H/T ) 3 configuration.

sophisticated models should be able to overcome all of the above-mentioned problems. 4.7. Influence of Working Conditions and System Properties on PeG. Most of the experiments aimed at characterizing the gas phase were carried out in the H/T ) 3 tank configuration. The raw data for demineralized water show that DeG becomes almost independent of the gas flow rate and impeller speed for tip impeller speeds higher than 3 m s-1 (provided that UG e 0.02 m s-1). It is also worth noting that the experimental DeG values are approximately in the same range as those calculated with the correlation given by van’t Riet and Tramper39 for bubble columns (for H/T < 3)sexcept for the lowest gas rate in which case the experimental values are 3-4 times greater than those calculated with the mentioned correlation. The data are better analyzed in terms of the dimensionless parameter PeG. The values for the coalescent system are plotted in Figure 11, where πND rather than N is used because the tip speed was found to be the relevant scale-up criterion for gas-phase behavior with Rushton turbines.19 At the smallest gas rate, PeG decreases with N, which is consistent with one’s expectation and confirms previous findings for the Rushton turbines.19,21 Only at the maximum investigated impeller speed does PeG increase with the gas flow rate. In all of the other conditions, intersections of the lines that interpolate the data are apparent. If the influence of the gas flow rate on PeG is neglected (for tip speeds greater than 3 m s-1), its value is in the range 3-9 for H/T ) 3. For comparison, the average correlating line found with the Rushton turbines19 is also drawn in the plot: overall, the present data are lower, which is explained with the higher circulation provided by these impellers. It is interesting to note that most of the slope changes in the correlating lines of Figure 11 can be explained with transitions in the bulk flow patterns identified for these systems. For example, for the lowest gas rate, the asymmetric regime prevails at the minimum impeller speed; it switches to the competitive and then to the complete dispersion regimes when N is increased, with a consequent improvement in gas macromixing. For the highest gas rate, PeG is essentially constant at low impeller speed, and then it increases when N > 7.4 s-1 on occurrence of complete dispersion. The behavior of the gas in the H/T ) 2 configuration is quite similar to that in the H/T ) 3 geometry. The main difference is in the lower PeG values (namely, 3-6 for intermediate to high impeller speeds). In addition,

the PeG changes with impeller speed are slightly more limited with respect to those of the previous case. Experiments were also performed with liquids other than demineralized water, namely, a 0.2 M sodium sulfate solution and a PVP solution. Qualitatively, the results are similar to those obtained with the Rushton turbines and the same liquids.19 For the sulfate solution, the fit between the experimental and the theoretical curves is slightly worse than that with water, especially at low gas rate, but still acceptable. The influence of N and QG on PeG is similar to that of demineralized water; see Figure 12. The PeG values, however, are lower (about 50%) at any condition in either H/T ) 3 or H/T ) 2 configurations. This reduction is a consequence of the smaller bubble size typical of the noncoalescent systems: they are entrained more easily in the circulating liquid, thus giving rise to enhanced gas recirculation, higher holdup,40 and more intensive gas macromixing. As a limiting condition, few experiments were also conducted with the noncoalescent solution at nil impeller speed and two gas rates. Gas dispersion in these cases was rather poor, being characterized by heterogeneous flow and highly inhomogeneous holdup with predominance of big bubbles ascending mainly close to the shaft. As a consequence of uneven bubble distribution and bubble velocity across each section (as well as less regular gas withdrawal) which result in the difficulty in getting a representative average gas composition from the sample, the measurements performed in these conditions are much less reliable. Qualitatively, though, the shape of these curves suggests a behavior closer to plug flowsconsistent with the trend of the data obtained under regular operating conditions. With the viscous PVP solution, the agreement between the experimental and the model curves is definitely worse than that with the other liquids: the experimental RTD curves are much less reproducible and, on average, are characterized by a narrower and slightly anticipated peak with respect to the theoretical ones. This last behavior is identical with that obtained with Rushton turbines19 and can be attributed to the nearly bimodal bubble-size distribution:5,34,35 while the smallest bubbles are easily entrained in the circulating liquid, the bigger ones reside in the tank for a shorter time.13 The model is not adequate, therefore, to describe these systems; no attempt was made to interpret these conditions with a more reliable one. If, nevertheless, the mentioned inaccuracy is disregarded and the experimental curves are still interpreted with this model, it

Ind. Eng. Chem. Res., Vol. 39, No. 9, 2000 3209 Table 1. Mean PeG Values for Multiple-Impeller Systems and Liquidsa liquid

3A315

2A315

3R

2R

water sodium sulfate PeG ratio (water-sodium sulfate)

6 3.7 1.6

4.5 2.5 1.8

10 6 1.7

6.7

a Data on Rushton turbines (2R and 3R): from ref 19. All data: average values at tip speed > 3 m s-1 (for all gas rates).

is seen that PeG values for N > 6 s-1 are about 1. This value is in reasonable agreement with those obtained by Manikowski et al.13 with cmc solutions. 4.8. Comparison of the Gas Behavior with Different Systems and Impellers. The average values obtained in this investigation with water and the sulfate solution and with H/T ) 3 or H/T ) 2 at a tip speed greater than 3 m s-1 are shown in Table 1. For the sake of comparison, similar data determined previously with multiple Rushton turbines19 are included. As is apparent, the ratio of PeG values for coalescent and noncoalescent liquids is nearly constant (about 1,7) regardless of tank configuration and agitator type: this fact suggests that the impeller coalescence rate is similar for the two sets of impellers and confirms the relevance of liquid properties in determining the bubble size and overall gas mixing. The gas macromixing state, as measured by PeG, is physically dependent on the extent of bubble entrainment by the liquid, which depends primarily, in turn, on liquid circulation, velocity profiles, and bubble size. It is noted that for a given liquid the PeG ratio for the multiple Rushton turbines and for the multiple hydrofoil impellers is in the range 1.5-1.7 (Table 1): these values are compatible with both “energetic” and “circulation” models.41,42 The analysis of data from this point of view, however, is beyond the scope of this paper. In the mentioned investigation on the Rushton turbines,19 it was found that the influence of the number of turbines can be described with the empirical equation

PeG(nj) ) PeG(ni)

() nj ni

nj/ni

(13)

where nj and ni are consecutive numbers. The ratio nj/ ni was taken to be equal to the aspect ratio H/T, and the adjusting function (nj/ni)nj/ni accounts for the declining influence of the number of impellers at their increase. For the pairs nj ) 3 and ni ) 2, the adjusting function is equal to 1.84. The values of PeG(3) and PeG(2) obtained in this investigation under equal operating conditions were compared with those calculated with eq 13: the hydrofoil data are slightly lower than those calculated with the equation. This corresponds to a slightly higher exponent in the adjusting function (about 1.7 instead of 1.5) and suggests higher impeller interaction, which is consistent with the axial flow feature of these impellers. 5. Conclusions The paper deals with the study of liquid and gas behavior in tanks sparged and stirred with multiple impellers of one specific style (pumping downward). The investigation was carried out in tanks of two aspect ratios and with liquids of different properties. As

regards the liquid-phase behavior, the main results can be summarized as follows: (a) The liquid macromixing state can be interpreted with the simple axial dispersion model, with the dispersion parameter being essentially independent of the measuring axial and radial position. On the contrary, the cascade of ideal stages with backmixing fails to interpret the liquid behavior satisfactorily. (b) As for the influence of the main operating parameters, aeration reduces the liquid-phase axial dispersion coefficientsat least with low-viscosity liquids. The axial dispersion coefficient is greatly affected by the impeller speed, provided that the aeration rate and viscosity are not too high. If these last conditions prevail, the dispersion coefficient is reduced by the gas rate, while on high aeration the gas influence becomes almost nil. On the contrary, with liquids of high viscosity, the dispersion coefficient on aeration is even higher than that obtained in the absence of gas. For the gas phase, the following features have been determined: (a) With the air-water system, the experimental RTD curves are fitted well with the theoretical ones provided by the axial dispersion model, which can therefore be confidently used for describing the gas behavior. (b) The influence of the impeller speed on the model parameter, PeG, is definite enough, while the effect of the gas flow rate is rather erratic, being linked to the combination of the flow regimes of each impeller. (c) The axial dispersion model is also acceptable with the electrolyte, coalescence-inhibiting solution, in which case roughly 40% lower PeG values were obtained. The same model, however, proves poor for describing the gas behavior with the viscous liquid. Although the above-mentioned results are relative to a specific impeller type, it is believed that the general behavior is representative of multiple hydrofoil impellers at largesat least for fluid dynamic model identification and in terms of operating parameter influence. In fact, such a generalization has already been shown to work for the liquid phase under unaerated conditions.32 For both phases, the axial dispersion model is acceptable for describing the behavior of this equipmentsat least to a first approximation. Empirical correlations of the dimensionless parameterssi.e., DeL/ND2 and PeG for the liquid and gas phases, respectivelyshave been proposed that can be used for preliminary process estimates. In fact, scrutiny of the experimental data has revealed a few specific weak points for this simple model which, with the documented tridimensional bubble motion13 and intermittent gas flow at the highest gas flow rates, call for additional study and more sophisticated modeling. Acknowledgment The authors thank Lightnin Mixers Equipment (Rochester, NY) for the use of their hydrofoil impellers during this investigation. The work was financially supported by University of Bologna and Italian Ministry of University and Research. The collaboration of Ms. D. Benfenati, Ms. C. Farolfi, and Messrs. L. Lodi, S. Pasqui, and V. Venturi in carrying out the experimental program is gratefully acknowledged. Part of the gasphase data were presented at NAMF Mixing XV Conference (Banff, June 18-23, 1995) and 13th CHISA Congress (Prague, Aug 25-27, 1998).

3210

Ind. Eng. Chem. Res., Vol. 39, No. 9, 2000

Nomenclature A1, A2 ) constants in eqs 10 and 11 a ) exponent in eq 11 cL ) tracer concentration in the liquid in general (kmol/ m3) cj ) liquid concentration in stage j of the CISB model (kmol/ m3) CG ) dimensionless tracer concentration in the gas CL ) dimensionless tracer concentration in the liquid D ) turbine diameter (m) dB ) bubble size (mm) DeG ) axial dispersion coefficient for the gas (m2/s) DeL ) axial dispersion coefficient for the liquid (m2/s) DS ) gas sparger diameter (m) f ) backflow rate (m3/s) Fl ) QG/ND3, flow number Fr ) N2D/g, Froude number g ) gravitational acceleration (m/s2) H ) liquid height (m) H ) Henry constant (Pa) j ) generic stage in the CISB model J ) total number of stages in the CISB model K ) (KLaH)/(mUG), dimensionless mass-transfer coefficient KLa ) volumetric mass-transfer coefficient (s-1) m ) H/RTCsolvent, partition coefficient N ) rotational speed (s-1) ni, nj ) number of impellers n0 ) tracer moles injected into the tank in the dynamic experiments (kmol) PeG ) (UGH)/(DeG), gas Pe´clet number Pg ) power draw under gassed conditions (W) Pu ) power draw under ungassed conditions (W) QG ) gas supply rate (m3/s) r ) radial coordinate (m) Re ) ND2F/η, Reynolds number S ) tank cross-sectional area (m2) t ) time (s) T ) tank diameter (m) t95 ) mixing time for 95% homogeneity (s) tR ) H/UG, gas mean holding time (s) UG ) QG/S, superficial gas velocity (m/s) V ) liquid volume in the tank (m3) z ) axial coordinate δ ) exponent in eq 10 δ(t) ) Dirac function (s)  ) fractional gas holdup ζ ) z/H, dimensionless axial coordinate η ) liquid viscosity (mPa‚s) ηref ) reference liquid viscosity, equal to 1 mPa‚s θ ) dimensionless time Subscripts j ) referring to stage j (CISB model) J ) referring to stage J (CISB model)

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Received for review February 14, 2000 Revised manuscript received April 20, 2000 Accepted May 24, 2000 IE000216+

ADDITIONS AND CORRECTIONS

Volume 39, Number 9 Analysis of the Fluid Dynamic Behavior of the Liquid and Gas Phases in Reactors Stirred with Multiple Hydrofoil Impellers. D. Pinelli and F. Magelli* Page 3202. The correct form of eq 6 is

-DeL

|

∂cL n0 ∂z z)0 ) S δ(t)

and

|

∂cL )0 ∂z z)H

(6)

instead of

cL|z)0 - DeL

|

∂cL ) n0δ(t) ∂z z)0

IE991088W 10.1021/ie991088w Published on Web 09/05/2000

and

|

∂cL ∂z z)H ) 0 (6)