Ind. Eng. Chem. Res. 2007, 46, 5021-5031
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Numerical and Experimental Study of a Dual-Shaft Coaxial Mixer with Viscous Fluids Maya Farhat, Christian Rivera, Louis Fradette, Mourad Heniche, and Philippe A. Tanguy* URPEI, Department of Chemical Engineering, EÄ cole Polytechnique Montreal, P.O. Box 6079, Station Centre-Ville, Montre´ al, Quebec H3C 3A7, Canada
The performance of a coaxial mixer equipped with an anchor combined to either a radial or an axial impeller was investigated. The system was operated in both counter- and corotating modes in the laminar and transition flow regimes. Experimental and numerical data regarding power consumption, mixing time, tracer evolution, and flow characterization were collected and compared. On the basis of these mixing efficiency criteria, the corotating mode was found to be a better choice for both mixers in the investigated flow regimes. The axial impeller-anchor mixer was also found to be more efficient in terms of energy consumption and mixing energy compared to the Rushton-anchor configuration. 1. Introduction Mixing operation is used in numerous industrial applications such as thermal homogenization, emulsification, fermentation, and polymerization. Unfortunately, a universal mixing system does not exist for all these applications, and the design of a mixing system as well as its optimization is most often application-related. Several factors can affect the effectiveness of a mixing system, namely, the type of agitator, the flow regime, and the product formulation. Many agitators have been developed in the last few years to handle viscous and non-Newtonian problems encountered in the industry:1 planetary mixer; off-centered double agitator; and coaxial mixer mounted with various agitator geometries, rotating at different speeds and operated in either corotating or counterrotating modes. In industrial applications, coaxial mixers are gaining in popularity because of their flexibility and good efficiency2 when the viscosity changes significantly during the process time. Although recent research work has shed some light on the performance of these mixers, their design is still mainly based on empirical knowledge, and research is needed to find the best way to optimally design, operate, and scale up this versatile mixer. In 2003, Ko¨hler and Hemmerle3 studied a coaxial system consisting of a double-pitched blade turbine coupled with an anchor in the counterrotating mode only. They investigated the power characteristics of this system in the transitional and turbulent regimes. They concluded that the speed ratio between the inner and the outer agitator has a significant influence on the power draw. In 2004, Foucault et al.2 studied three different dispersion disks, namely, a Deflo impeller, a Sevin impeller, and a Deflo-Sevin hybrid impeller. They found that the hybrid impeller-anchor combination was the most efficient for mixing in both rotating modes (co- and counterrotating), irrespective of the fluid rheology. They also studied minimum agitation conditions to achieve the just-suspended state of solid particles (Njs), and they determined that Njs had lower values with the impellers having the best axial pumping capabilities. In 2005, Foucault et al.4 established new correlations based on impeller geometry for the generalized Reynolds number and the power * Corresponding author. E-mail:
[email protected]. Fax: 514-340-4105.
Figure 1. Experimental setup.
number, and they extended the concept of power master curve to coaxial mixers. Mixing-time and power-consumption master curves were obtained for Newtonian and non-Newtonian fluids based on a modified Reynolds number that takes into account the particular rotational configuration of the anchor-turbine in the coaxial system. They showed that the corotating mode was better than the counterrotating mode in terms of mixing time. In 2006, Foucault et al.5 compared the performance of a Rushton turbine, a Sevin impeller, and a Deflo-Sevin dispersing disk. They concluded that the Rushton turbine was the most effective one in terms of homogenization. Also in 2006, Rudolph et al.6 used the Foucault et al. correlations and a single master-curve technique to study a coaxial mixer consisting of a dual set of A200 impellers and an anchor impeller in the corotating mode. Working in the laminar-to-turbulent regime, they showed that the A200 is not affected by the speed of the anchor but that, in contrast, the close clearance impeller is affected by the speed of the A200. Recently, Rivera et al.7 examined the origin of the superior performance of the corotating mode by means of three-dimensional, finite-element flow simulations. It was found that the reduced pumping action and the flow compartmentalization of the counterrotating mode explain why the corotating operation is a better choice for the homogenization of viscous products.
10.1021/ie061226z CCC: $37.00 © 2007 American Chemical Society Published on Web 06/07/2007
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Figure 2. Agitators used in the work: (a) Rushton turbine and (b) Mixel TT.
The torquemeter range for the central turbine (Himmeslstein Inc.) is 0-22.6 N‚m (precision of (0.1% full scale). The one for the anchor (Vibrac Corp.) is 0-11.3 N‚m (precision of (0.5% full scale). 2.2. Numerical Method. To predict numerically the unsteady three-dimensional flow field in a stirred tank, the incompressible Navier-Stokes equations were solved using the finite-element method. Figure 3. Specific dimensions for 20 cm Rushton turbine.
The literature on coaxial mixing focuses mainly on radial impellers and dispersing disks. Not much is known regarding the behavior of axial impellers, in particular, whether the corotating mode is still a preferred mode of operation. The objective of this work is to examine this issue and highlight the performance differences between a coaxial mixer provided with a radial-flow impeller (Rushton turbine) and a coaxial mixer with an axial-flow impeller (Mixel TT). In order to achieve this objective, a dual methodology based on an experimental and numerical approach will be followed. 2. Materials and Methods 2.1. Experimental Setup and Methodology. A fully instrumented coaxial mixer of 46 L shown in Figure 1 was used for the experiments. It consists of an impeller and an anchor mounted on two independently actuated coaxial shafts running respectively at high and low speed. The rig also includes three J-type thermocouples (iron-constantan junction) of very weak thermal inertia (naked thermocouple, deprived of any stainless sheath). The agitators considered are a radial turbine (Rushton) (Figure 2a) and an axial impeller (Mixel TT) in pumping-down mode (Figure 2b). Both agitators have a diameter of 20 cm. Specific dimensions for the Rushton turbine are presented in Figure 3. For the Mixel TT (patented design), the projected height is 5 cm and the projected width of each profiled blade is 6.5 cm. The rig can be operated in corotating or counterrotating mode. Aqueous solutions of corn syrup were used for both combinations of anchor and impeller. The rheological properties of the fluids were determined with a Bohlin Viscometer 88-BV with Couette configuration. The viscosity of the corn syrup solutions was in the range of 0.1-35 Pa‚s. The density was 1350 kg/m3. Corotating and counterrotating modes with speed ratios (anchor/open impeller) of 1:10 and 1:30 were investigated. These operating conditions were selected in order to avoid the formation of a vortex at the free surface. The power draw of each impeller was determined from the torque measurements given by the torquemeters installed on each shaft (Figure 1).
F
(∂v∂t + v‚∇v) + ∇p + ∇‚τ ) f
(1)
∇‚v ) 0
(2)
A Galerkin finite-element method was used to solve the above equations. The locally mass conserving tetrahedral nine-node finite element P1+-P010 that approximates the velocity with a superlinear polynomial and considers a constant pressure was employed for the discretization. A domain-embedded method (virtual finite-element method or VFEM) was used to handle the moving impellers. It is useful to mention that, within the VFEM framework, a unique mesh is required during the computation and the impeller is discretized by moving control points that act as kinematics constraints through Lagrange multipliers on the governing Navier-Stokes equations. Figure 4 shows an example of the surface grids used to generate the control points representing the agitators (virtual objects). The impellers used in this work are refined versions of the ones presented in Figure 4, yielding about 3 100 and 3 400 control points for the Rushton and Mixel impellers, respectively. The reader is referred to papers by Glowinski8 and Bertrand et al.9 for further details about the domain-embedded method. The set of boundary and virtual conditions employed are expressed as follows: • No normal velocity at the fluid surface (vz ) 0); • No slip condition at the vessel wall (v ) 0); and • Imposed velocity on the control points v(t) ) ω × r(t). The nonlinear algebraic system arising from the discretization of the equations of change was resolved by a Newton-Raphson iterative scheme. For each set of variables, namely, the mixed velocity-pressure variables and the Lagrange multipliers variables, appropriate solution methods were selected, namely, the velocity-pressure variables were computed by the ILU preconditioned TFQMR method that belongs to the class of Krylov linear iterative solvers,10 and the Lagrange multipliers were computed by the Uzawa algorithm. The coupling between theses two sets of variables was made explicit. To take into account the unsteady nature of the flow, a Gear scheme was used. Computations were carried out until fully periodic solutions were obtained.
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Figure 4. Virtual impellers employed in this work: (a) Rushton-anchor impeller and (b) Mixel TT-anchor impeller.
From the authors’ experience, turbulence models are suitable to simulate mixing processes in the fully turbulent regime (Re . 1 000), but for stirred tanks operating in the lower part of the transition regime (Re < 1 000), it is much better in terms of accuracy to resort to small grid sizes, as has been done in this work and others such as that by Bartels et al.13 The simulations were performed with commercial 3D finiteelement software POLY3D (Rheosoft, Inc.) developed in our group. The total memory requirement was approximately 2 GB. All simulations were run on an IBM p690 computer cluster. The visual postprocessing of the results was carried out with Ensight software from CEI. The solutions were considered converged when the maximum relative error for the NewtonRaphson scheme was 7.6. The fluid viscosity measurements as well as the mixing experiments were performed at room temperature (∼23 °C). At the beginning of each experiment, the NaOH solution was added in order to turn the solution to purple. Then, a solution of HCl was added to change the color from purple to yellow. This color change
Figure 7. Planes employed to compute the pumping rate in the coaxial mixer.
was timed using a chronometer. The acidic and basic solutions were first mixed with 500 mL of the glucose-water solution in order to not change the density of the tracer. Also, in order to be consistent, the injection of the tracer was done at the same spot (near the central shaft). Each mixing-time measurement was repeated three times, and the results of the three tests were averaged in order to get tm.
Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007 5025 Table 1. Power Consumption of the Investigated Scenarios for the Rushton-Anchor Coaxial System
rotation mode
viscosity (Pa‚s)
speed HS-LS (rpm)
Re coaxial mixer
Pturbine Exp. (W)
Panchor Exp. (W)
Ptot/V Exp. (W/m3)
Ptot/V Num. (W/m3)
difference Ptot/V (%)
mixing time (s)
Ntm
corotating counterrotating single turbine corotating counterrotating single turbine corotating counterrotating single turbine
10 10 10 10 10 10 1 1 1
150-5 150-5 150-0 200-20 200-20 200-0 150-5 150-5 150-0
13.05 13.95 13.50 16.20 20.00 18.00 130.50 139.50 135.00
43.5 52.0 48.5 76.4 82.8 97.6 22.3 25.5 25.0
0.5 1.5 0 5.2 16.3 0 0.2 2.0 0
956.52 1163.04 1054.35 1773.91 2154.35 2121.74 489.13 597.83 543.48
1065.22 1117.39 1070.87 1760.86 2347.82 2110.08 383.48 435.98 410.38
11.36 3.93 1.57 0.74 8.99 0.55 21.60 27.07 24.49
184 3581 486 125 1568 281 31 253 59
445 9251 1215 375 5749 937 75 654 148
Table 2. Power Consumption of the Investigated Scenarios for the Mixel TT-Anchor Coaxial System
rotation mode
viscosity (Pa‚s)
speed HS-LS (rpm)
Re coaxial mixer
Pturbine Exp. (W)
Panchor Exp. (W)
Ptot/V Exp. (W/m3)
Ptot/V Num. (W/m3)
difference Ptot/V (%)
mixing time (s)
Ntm
corotating counterrotating single turbine corotating counterrotating single turbine corotating counterrotating single turbine
10 10 10 10 10 10 1 1 1
150-5 150-5 150-0 200-20 200-20 200-0 150-5 150-5 150-0
13.05 13.95 13.50 16.20 20.00 18.00 130.50 139.50 135.00
23.6 25.7 24.4 50.1 51.5 46.4 9.7 10.0 11.8
0.6 2.3 0 7.1 20.0 0 0.1 0.6 0
526.09 608.70 530.43 1243.48 1554.34 1008.70 213.04 230.43 256.52
556.70 587.72 556.14 986.36 1393.49 1074.7 224.00 234.76 255.32
5.82 3.45 4.85 20.68 10.35 6.54 5.14 1.88 0.47
240 428 252 90 180 152 30 55 40
580 1106 630 270 660 507 72.5 142 100
2.5. Tracer Distribution Evolution. To evaluate numerically the distributive performance of the studied configurations, the evolution of the position of massless nondiffusive tracers was tracked using an element-by-element approach.14 The tracers were initially positioned at the top of the tank. Their volume occupied (V) at each time step was considered as the volume of the elements with at least one tracer inside them. Each element volume was extracted from a reference mesh with a ratio between the number of tetrahedral elements and particles tracers close to 1. As a reference mesh, a rectangular box was preferred over a tanklike shape because the rectangular box mesh allows for having a more homogeneous distribution of tetrahedral volumes. The rectangular box was built with height and width equal to the mixer’s height and diameter, respectively. The number of tracers employed for the present work was ∼3 000. Figure 5 presents the reference mesh employed to quantify the tracer’s volume.
impellers’ volume. The limit tracer’s Volume (Vlim) value depends on the reference mesh and the number of tracers employed. 2.6. Numerical Flow Characterization. Pumping rate, shear strain rate, and vorticity circulation were computed over 8 planes composed of 2 596 triangular elements at heights of 0.12, 0.16, 0.20, 0.24, 0.28, 0.32, 0.36, and 0.4 meters from the vessel bottom. Figure 7 presents the planes that were used to compute these parameters. Pumping rate was used to identify important flow characteristics such as flow dominant orientation and magnitude. The computation of its total contribution in the radial (r) and tangential (θ) directions was performed by solving numerically the stream function (ψ). Then, the pumping rates in the radial and tangential directions were calculated by
Qr(z) ) (ψrmax - ψrmin)
(6)
One of the main drawbacks of this method is the nondiffusive character of the tracers, because if segregated zones in the flow exist, there is a good chance that the tracers will fall into a segregated zone and stay there indefinitely. Thus, for conditions where segregation can be significant, it is impossible to determine an eventual mixing time. However, this approach is still interesting since it allows the determination of the tracer’s distribution evolution, which is related to the mixing pattern of the studied scenarios under the hypothesis that the computed velocity field and trajectories are accurate enough. Furthermore, it gives a three-dimensional picture of the tracer’s distribution evolution in the vessel, which allows for identifying important performance phenomena such as flow compartmentalization and contrasting the distributive performance of industrial mixers in different conditions (see Figure 6).
Qθ(z) ) (ψθmax - ψθmin)
(7)
It was observed that it is very difficult to reach 100% of the volume occupied by tracers. For that, the limit tracer’s Volume (Vlim), which is defined as the maximum volume that the tracers can occupy for any of the studied conditions, was used. This value was set constant for all the conditions to 75% of the net volume of the tank ) total tank volume - total
In addition, the axial pumping rate was computed by means of
Qz(z) )
∫A v+z dA ) - ∫A v-z dA
(8)
For analysis purposes, the dimensionless flow numbers in the radial, tangential, and axial directions are helpful and were determined by
Nqi )
Qi NturbineD3
i ) r, θ, z
(9)
The magnitude of the shear strain rate gives us information about the amount of deformation inside the vessel. It is defined by
||γ˘ || )
x
(
)
∂vx 2 ∂vy 2 ∂vz 2 1 ∂vx ∂vy 2 + + + + + dx dy dz 2 ∂y ∂x 1 ∂vx ∂vz 2 1 ∂vy ∂vz 2 + + + 2 ∂z ∂x 2 ∂z ∂y
(
) (
)
(10)
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Figure 8. Comparison of the experimental power curves for the Rushton-anchor mixer and the Mixel-anchor mixer in the corotating mode, the counterrotating mode, and the turbine alone mode. Table 3. Parameters of the Moo-Young Correlation for the Investigated Coaxial Configurations coaxial system Rushton-anchor Rushton-anchor Rushton alone Rushton alone12 Mixel_TT-anchor Mixel_TT-anchor Mixel alone
Table 4. Specific Power for the Compared Scenarios
operation mode
A
a
corotating counterrotating anchor fixed literature values; unbaffled, no anchor corotating counterrotating anchor fixed
3 300 300 000 13 000 17 500
-0.78 -1.32 -0.91 -0.75
1 295 10 390 4 447
-0.59 -0.85 -0.75
The average of its value was computed over each plane. Finally, the vorticity circulation is interpreted as an approximate measure of the swirl intensity caused by the velocity gradients generated in the tank. It was computed with the help of the Stokes theorem by
ω)
∫(∇xv)‚n dA
(11)
3. Results 3.1. Power Consumption. Table 1 and Table 2 compare the experimental and numerical power numbers for several values of Reynolds number for Rushton-anchor and Mixel-anchor coaxial systems, respectively. The discrepancy between experimental and numerical data is very low except in a few cases, which can be explained by the uncertainty of the torque measurement for low torque values. Also, in a few cases for the Rushton-anchor configuration, the discrepancy can be explained by an underprediction of the numerical velocity gradients generated near the turbine blades at Reynolds numbers > 100. For the anchor, Tables 1 and 2 clearly show that the anchor consumes much less power than the turbine. In fact, because of the chosen speed ratios, the central turbine (high speed) consumes most of the power because of its high speed while the anchor power consumption, although pretty low, is mainly related to its shape and diameter (drag effect). Consequently, total consumed power is more affected by the central turbine (Rushton or Mixel) than by the anchor.
rotation mode
coaxial mixer configuration
P/VNum. (W/m3)
corotating counterrotating single turbine corotating counterrotating single turbine
Rushton-anchor Rushton-anchor Rushton-anchor Mixel TT-anchor Mixel TT-anchor Mixel TT-anchor
1 065 1 117 1 071 986 1 393 1 075
Figure 8 shows that the Mixel-anchor mixer required less power at the same conditions than the Rushton-anchor mixer, confirming the usual trend with axial-flow impellers. The reader should pay attention, in that figure, to the fact that the power number curves can be misleading. In fact, by looking at the curve for the corotating mode, one can think that this rotation mode consumes much more power than the other two modes (counterrotating and turbine alone), when, in fact, this is not the case. In the corotating case, the power number is higher but the power consumption (P) is lower than those in the other two modes. The definitions of the Reynolds number and the power number in this mode shift the Np number while decreasing the Re number. This is explained by the subtraction of the rotating speeds of each agitator in the definitions of Np and Re. 3.2. Mixing Time. Figure 9 presents the experimental dimensionless mixing time as a function of the Reynolds number for the Rushton-anchor mixer (Figure 9a) and for the Mixelanchor mixer (Figure 9b). Corotating and counterrotating modes are both shown on those figures. The mixing-time curves clearly demonstrate the superiority of the corotating mode over the counterrotating mode from a fluid-circulation point of view. It is worth mentioning that the dimensionless mixing times for large Reynolds numbers (Re > 1000) were constant and slightly smaller in the case of the Mixel-anchor mixer (Ntm ) 23) than for the ones obtained for the Rushton-anchor mixer (Ntm ) 25). The Moo-Young correlation Ntm ) A‚Rea was employed to quantify this trend.15 Table 3 gives the parameters of the correlation. As can be observed, the dimensionless mixing time for the Rushton-based system is more sensitive to Reynolds number changes since the Reynolds number exponents (a parameter)
Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007 5027
Figure 9. Experimental mixing-time curve for the coaxial mixer in (a) Rushton-anchor system and (b) Mixel-anchor system.
Figure 10. Radial pumping for the Rushton-anchor system.
Figure 12. Axial pumping for the Rushton-anchor system.
Figure 11. Tangential pumping for the Rushton-anchor system.
Figure 13. Radial pumping for the Mixel-anchor system.
are larger than the ones obtained for the Mixel-anchor system irrespective of the operation mode. Also, the experimental parameters for the Moo-Young correlation for the Rushton alone mode were found to be close to those from the literature in an unbaffled scenario.12 Unfortunately, no parameters were found in the literature for a Mixel TT system in the laminartransitional regime in order to be compared with the experimental ones. 3.3. Flow Patterns Description. The objective of this section is to present the effect of the interaction anchor-turbine over the obtained hydrodynamics for both studied configurations. Direct comparison between the Mixel-anchor and Rushton-
anchor are difficult because of the fact that they consume different amounts of energy. In spite of this, some differences in the main flow pattern produced by these two different mixers will be exposed. Figures 10-15 present the radial, tangential, and axial circulations for the Rushton-anchor and Mixelanchor coaxial systems, respectively, at different heights. Turbine alone, corotating, and counterrotating modes are shown. The speeds of the turbine and anchor are set to 150 and 5 rpm, respectively. The fluid viscosity was 1 Pa s (Re ) 135). From these plots, the positive effect of the anchor over both systems can be observed when operating in the corotating mode because of an increase in both tangential and axial circulations in the
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Figure 14. Tangential pumping for the Mixel-anchor system.
Figure 15. Axial pumping for the Mixel-anchor system.
Figure 17. Vorticity circulation for the Mixel-anchor system.
Figure 18. Average shear strain rate for the Rushton-anchor system.
Figure 16. Vorticity circulation for the Rushton-anchor system. Figure 19. Average shear strain rate for the Mixel-anchor system.
zones located above and below the turbine with respect to the data obtained for the single turbine. Therefore, the anchor in corotating mode promotes two main effects: it induces a larger pseudo-cavern around the turbine and intensifies the upwarddownward motion. Nevertheless, if the anchor and turbine rotate in opposite ways, the situation is reversed. In this case, the radial pumping is the discharge direction that is intensified in comparison with the single-turbine data (Figures 10 and 13). This reduction of tangential and axial motions for counterrotating mode is related to the formation of secondary circulation regions around the anchor that shrinks the size of the pseudo-cavern close to the turbine, enhancing the radial component of the flow. Finally, if
the turbine is rotating and the anchor remains fixed, we obtain a flow that is in between those for the corotating and counterrotating modes, as the presented pumping profiles have shown. That would explain the average performance observed in the mixing-time experiments. It is worth highlighting the pattern observed in Figure 12, where two separated zones of axial pumping, one above and the other below the impeller, are easily identified irrespective of the rotating mode. For the case of the Mixel-anchor mixer, the axial-circulation barrier observed for the Rushton-anchor is nonexistent, as shown in Figure 15. This is explained by the radial discharge of the Rushton turbine, which acts as a barrier between these regions, promoting an axial-flow compartmen-
Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007 5029
Figure 20. Tangential pumping number for both geometries at similar power dissipation.
Figure 23. Tracers distribution evolution for the Rushton-anchor configuration.
Figure 21. Axial pumping number for both geometries at similar power dissipation. Figure 24. Tracer distribution evolution for the Mixel-anchor configuration.
Figure 22. Dimensionless shear strain rate for both geometries at similar power dissipation.
talization. For the case of the Mixel turbine, the profiled shape seems to minimize this effect, having a stronger impact over the hydrodynamics of the stirred tank. To better understand the pseudo-cavern contraction-expansion phenomenon, Figures 16 and 17 present the vorticity circulations for both mixers. Even if the counterrotating mode has a larger vorticity circulation at the turbine plane for the Rushton-anchor mixer, the swirling contracts as we move above the turbine. For the case of the Mixel-anchor mixer, a similar scenario is presented with the exception that the contraction is observed for both the upper and lower zones of the tank, as
Figure 25. Tracers distribution evolution difference between Mixel-anchor and Rushton-anchor configurations at similar specific power.
Figure 17 shows. Figures 18 and 19 illustrate the average shear strain rate for the studied conditions. In both configurations, the deformation is larger for the corotating mode in the area close to the turbine. However, this situation is inversed at the lower and upper zones of the tank, with the corotating mode being the one that offers the larger shear strain rate. This
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Figure 26. Experimental dimensionless mixing energy for the coaxial mixer in (a) Rushton-anchor system and (b) Mixel-anchor system.
supports the observations already mentioned about the effect of the anchor rotation over the flow generated in the mixer. 3.4. Circulation Patterns at Constant Specific Power. As was mentioned earlier, a direct comparison at constant Reynolds number between the Mixel-anchor and Rushton-anchor coaxial mixers is unfair because of different amounts of power that they consume. For that reason, in this section, our task is to present a direct comparison between the Mixel-anchor and Rushton-anchor mixers for the different rotating modes. The conditions and specific power for each compared case are displayed in Table 4. As can be seen, the values are pretty close to each other except for the counterrotating Mixel-based mixer that exhibits a considerable difference of power with respect to the other two. Because of the fact that, for this comparison, the Reynolds number becomes smaller than in the last section, it is more difficult to observe the anchor-turbine interactions, as all the plots show. For the assessment, dimensionless pumping numbers are employed to attenuate the fact that different speeds were employed for the Mixel-anchor and Rushton-anchor configurations. Figures 20 and 21 illustrate the differences in circulation for the investigated cases. It is observed that the Mixel-anchor system outperforms the Rushton-anchor mixer in overall flow pumping. It can be noted in Figure 20 that the maximum value of tangential pumping is not at the same axial location for the Mixel-anchor and the Rushton-anchor, being at the half of the Rushton turbine blade for the latter and at the top of the turbine for the former. This is caused by the profiled shape of the Mixel TT. This affects not only the tangential motion but the average shear strain rate, too, as Figure 22 shows, where the same phenomenon is observed. From this graph, it is worth remarking on the reduction in dimensionless shear strain rate caused by the addition of the Mixel TT in the coaxial system. Furthermore, from Figure 20, it can also be observed that, for the case of the Rushton-anchor, the tangential motion seems to reduce drastically in the upper zone of the vessels. For the Mixel-anchor configuration, this effect is attenuated. In relation to the axial motion, we can observe in Figure 21 for the Mixelanchor system an increase of axial movement with respect to the results offered by the Rushton-anchor vessel. This axial intensification goes from the turbine height to the top of the tank. 3.5. Tracer Distribution Evolution. To observe the effect of the described flow patterns over the distributive performance of the mixers, the tracer distribution evolution was compared. Figures 23 and 24 present the obtained curves for two different fluid viscosities. Figure 25 displays the obtained curves at similar power dissipation in reference to the cases studied in Section 3.4. As expected, corotating mode gives a faster and better tracer
distribution. Furthermore, it is noted that the volume occupied by the tracers for the Mixel-anchor system is slightly larger than the one obtained for the Rushton-anchor system in the majority of the presented scenarios. This is explained by the axial compartmentalization zone mentioned earlier that is present in the Rushton-anchor configuration, making it easier for the tracers to fill the vessel volume when they are agitated by the Mixel-anchor impellers. This trend can also be observed in the mixing-time data. 3.6. Mixing Energy. Finally, to assess the mixing effectiveness, the variation of the dimensionless mixing energy, Emix ) NpNtm, is shown versus the Reynolds number in Figure 26. The fact that the Mixel TT-anchor mixer consumes much less energy than the Rushton-anchor mixer to achieve the same result appears clearly, which makes the Mixel-anchor mixer more attractive for mixing operations where energy cost is an issue. Also, it is easy to conclude here again that the corotating mode consumes less energy than the two other modes. 4. Conclusion The objective of this work was to put in evidence the differences in the flows produced by two coaxial mixer configurations in counter- and corotating modes for viscous fluids. The configurations included an anchor rotating at low speed and either a radial impeller (Rushton turbine) or an axial impeller (Mixel TT), rotating at high speed while remaining in the laminar and transitional flow regimes. The numerical and experimental results both showed that the Mixel-anchor mixer requires less power than the Rushton-anchor mixer at comparable Reynolds numbers. Furthermore, the corotating mode proved once again to be more efficient than the counterrotating mode in terms of mixing time, as was found by the previous work of Foucault et al.2,4,5 Also, from experimental analysis, we concluded that the Mixel-anchor mixer also exhibits lower mixing time. Numerically, it was found that the anchor affects the mixing positively in the corotating mode for both coaxial mixers because of an increase in both tangential and axial circulation in the zones located above and below the turbine with respect to the data obtained for the single turbine. Therefore, the anchor in corotating mode promotes two main effects: it induces a larger pseudo-cavern around the turbine and intensifies the upward-downward motion. In the counterrotating mode, the radial pumping is the discharge direction that is intensified in comparison with the single-turbine data, and the reduction of tangential and axial motion is related to the formation of secondary circulation regions around the anchor that shrink the size of the pseudo-cavern close to the turbine, enhancing the radial component of the flow. From the numerical circulation-pattern results, it was observed that the Mixel-
Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007 5031
anchor system outperforms the Rushton-anchor mixer in overall flow pumping. It was also observed that, for the case of the Rushton-anchor mixer, the tangential motion seems to reduce drastically in the upper zone of the vessels. For the Mixelanchor configuration, this effect is attenuated and the axial intensification goes from the turbine height to the top of the tank. On the other hand, tracer distribution evolution confirmed, once again, that the corotating mode is superior since it gives a faster and better tracer distribution. Furthermore, this technique showed that the volume occupied by the tracers for the Mixelanchor system is slightly larger than the one obtained for the Rushton-anchor system, which can be explained by the axial compartmentalization zone mentioned earlier that is present in the Rushton-anchor configuration. Finally, on a mixing-energy basis, the Mixel TT-anchor mixer proved to have a higher performance than the Rushton-anchor mixer. Consequently, using the experimental and numerical results provided in this paper, it can be concluded that the pumping-down Mixel TTanchor mixer in corotating mode represents the most energyefficient mode of all the investigated scenarios. Hence, the Mixel TT-anchor coaxial configuration is more attractive for mixing operations where energy cost is an issue. Acknowledgment The support of NSERC and the research consortium on NonNewtonian Innovative Mixing Technologies is gratefully acknowledged. Nomenclature Cw ) gap between the anchor and the wall, m Emix ) mixing energy Da ) diameter of the anchor, m Dc ) diameter of the tank, m Dt ) diameter of the central turbine, m H ) fluid height in the mixing tank, m Hc ) height of the tank, m Kp ) power constant (Newtonian fluid) L ) width of the agitator blades, m M ) torque, N‚m Mc ) corrected torque, N‚m Mm ) measured torque, N‚m Mr ) residual torque, N‚m N ) rotational speed, rps n ) unit normal vector Na ) rotational speed of the anchor, rps Np ) power number Nq ) flow number Nq,r ) flow number, radial Nq,θ ) flow number, tangential Nq,z ) flow number, axial Nt ) rotational speed of the central turbine, rps P ) power, W P ) pressure, Pa Q ) pumping rate, m3/s Re ) Reynolds number RN ) speed ratio, Nt/Na T ) diameter of the mixing tank, m t ) time, s
tm ) mixing time, s V ) volume of the mixing tank, m3 V ) velocity, m/s W ) height of the agitator blades, m Wa ) anchor blade thickness, m γ˘ ) shear rate, s-1 µ ) viscosity, Pa‚s F ) density, kg/m3 τ ) stress tensor, Pa Ψ ) stream function ω ) vorticity circulation ∫f dA ) surface integration ∫f dΩ ) domain integration Literature Cited (1) Tanguy, P. A.; Thibault, F.; Brito-de la Fuente, E.; Espinosa-Solares, T.; Tecante, A. Mixing performance induced by coaxial flat blade-helical ribbon impellers rotating at different speeds. Chem. Eng. Sci. 1997, 52 (11), 1733-1741. (2) Foucault, S.; Ascanio, G.; Tanguy, P. A. Coaxial Mixer Hydrodynamics with Newtonian and non-Newtonian Fluids. Chem. Eng. Technol. 2004, 27 (3), 324-329. (3) Ko¨hler, S.; Hemmerle, W. Analysis of the power characteristic of a coaxial agitator with varied diameter and speed ratio inner and outer mixing device. 11th Eur. Conf. Mixing (Bamberg, Germany) 2003, 14-17. (4) Foucault, S.; Ascanio, G.; Tanguy, P. A. Power Characteristics in Coaxial Mixing: Newtonian and Non-Newtonian Fluids. Ind. Eng. Chem. Res. 2005, 44, 5036-5043. (5) Foucault, S.; Ascanio, G.; Tanguy, P. A. Mixing times in coaxial mixers with Newtonian and non-Newtonian fluids. Ind. Eng. Chem. Res. 2006, 45 (1), 352-359. (6) Rudolph, L.; Scha¨fer, M.; Atiemo-Obeng, V.; Kraume, M. High viscosity mixing: Experimental and numerical explorations of co-axial mixers. 12th Eur. Conf. Mixing (Bologna, Italy) 2006. (7) Rivera, C.; Foucault, S.; Heniche, M.; Espinosa-Solares, T.; Tanguy, P. A. Mixing analysis in a coaxial mixer. Chem. Eng. Sci. 2006, 61 (9), 2895-2907. (8) Glowinski, R. Numerical methods for fluids. In Handbook of numerical analysis; Ciarlet, P. G., Lions, J. L., Eds.; North-Holland: Amsterdam, The Netherlands, 2003; Vol. IX. (9) Bertrand, F.; Tanguy, P. A.; Thibault, F. A Three-Dimensional Fictitious Domain Method for Incompressible Fluid Flow Problems. Int. J. Numer. Methods Fluids 1997, 25, 719-736. (10) Freund, R. W. A transpose-free quasi-minimum residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Comput. 1993, 14, 470482. (11) Rivera, C.; Heniche, M.; Ascanio, G.; Tanguy, P. A. A virtual finite element model for centered and eccentric mixer configurations. Comput. Chem. Eng. 2004, 28, 2459-2468. (12) Brooks, A. N.; Hughes, T. J. R. Streamline upwind-Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 1982, 32, 199-259. (13) Bartels, C.; Breuer, K.; Wechsler, K.; Durst, F. Computational fluid dynamics applications on parallel-vector computers: Computations of stirred vessel flows. Comput. Fluids 2002, 31, 69-97. (14) Heniche, M.; Tanguy, P. A. A new element-by-element method for trajectory calculations with tetrahedral finite element meshes. Int. J. Numer. Methods Eng. 2006, 67 (9), 1290-1317. (15) Moo-Young, M.; Tichar, K.; Dullien, F.A.L. Blending efficiencies of some impellers in batch mixing. AIChE J. 1972, 18 (1), 178-182.
ReceiVed for reView September 18, 2006 ReVised manuscript receiVed December 11, 2006 Accepted April 18, 2007 IE061226Z
