Inclined-Shaft Agitation for Improved Viscous Mixing - American

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Inclined-Shaft Agitation for Improved Viscous Mixing Steven Wang,*,†,‡ Jie Wu,§ and Naoto Ohmura∥ ‡

CSIRO Earth Science and Resource Engineering, Clayton, Victoria 3168, Australia CSIRO Process Science and Engineering, Highett, Victoria 3190, Australia ∥ Department of Chemical Science and Engineering, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan † Department of Chemical Engineering, Monash University, Clayton, Victoria 3800, Australia §

ABSTRACT: The mixing of viscous Newtonian fluids was investigated in experiments by means of decolorization visualization. Various inclined shaft and offset impeller configurations were tested to develop complete mixing with less energy consumption. It was found that isolated mixing regions could be more easily destroyed with larger angles of inclination, for example, 30°. A specific energy parameter was introduced to quantify the energy required for complete homogenization based on the mixing time and the specific power input. It was found that an increased shaft angle reduces the specific energy required for complete mixing. The results suggested that operating an angled-shaft system at higher speeds is more energy-efficient. It was also found that increased eccentricity and reduced impeller-to-bottom clearance generally lead to reduced specific energy consumption for homogenization. The CFD results confirmed that breaking the spatial symmetry in an angled-shaft system leads to destruction of the regular regions in the mixing tank. In addition, the CFD simulations showed that liquid blending and particle dispersion can both be enhanced by using inclined-shaft systems.

1. INTRODUCTION Mixing of fluids is a critical process that occurs throughout modern industry in many chemical, hydrometallurgical, and biological processes. A fluids-mixing system can be operated in turbulent, transitional, or laminar flow regimes depending on the rheology of the materials involved. Operating a mixing system in a vessel in laminar conditions can be challenging at very high viscosities because of incomplete mixing, which usually leads to poor homogeneity of the materials in the vessel. Mechanically agitated tanks are commonly used in industry for blending liquids and dispersing solid particles throughout the tanks. In many cases, the liquid phase in industrial operations can have a significantly high viscosity because of the large quantities of chemical and organic substances present. Nagata et al.1 reported that homogenization is not easily accomplished in highly viscous liquids with conventional turbulent-type impellers. In their experiments, the decolorization method, using sodium thiosulphate and iodine, was emploted to evaluate the mixing times in highly viscous liquids. It was found that homogeneous mixing was not attained, which led to large variations in material concentrations throughout the system. In Newtonian flow systems, the passing of the impeller blades triggers the onset of chaos by introducing small perturbations in the underlying regular two-dimensional flow that is observed when impellers are replaced by disks. However, the doughnut-shaped regular regions are still present above and below the impeller.2−4 Isolated mixing regions (IMRs) refer to confined mixing regions, segregated by well-defined boundary layers. IMRs in a stirred vessel are commonly encountered in the form of toroidal vortices in laminar flows at low Reynolds numbers.2,4 The structural properties of IMRs are highly complex, consisting of several filaments and a core torus.3−5 Makino et al.5 stated that the geometric structure of IMRs © 2013 American Chemical Society

could be affected by the number of blades and the shaft speed. So far, the most challenging aspect associated with the mixing of viscous fluids in a stirred tank is the formation of such complex isolated mixing regions, which is a great barrier for mixing purpose. Running the agitator system at high speeds could solve the IMR problem.7 However, considering the substantial power input into mixing vessels, it is often not practical to improve the mixing by increasing the impeller speed. Unsteady and timeperiodic speeds have also been introduced, with some success, to enhance the chaotic degree in a mixing vessel.8,9 The enhancement in chaotic degree eventually improves the homogeneity of the material through the system by completely destroying the isolated mixing regions. However, the drawback with these methods is that they are highly impractical for largescale mixing tanks used in hydrometallurgical processes, where steady rotation is always favorable. Other measures aimed at improving laminar mixing include the installation of baffles, the installation of off-center impeller(s), and the insertion of large objects.6,10,11 Although these approaches markedly enhance the mixing performance, further attempts to improve the mixing quality in a mixing vessel under laminar conditions should be employed in response to the needs required to shorten the mixing time and reduce the energy input into the systems. Recently, Takahashi et al.12 investigated the mixing performance in a vessel with an inclined impeller. They reported that this mixing system does not cost as much because a traditional impeller such as a Rushton turbine is inserted into the vessel. They confirmed the effectiveness of the impeller inclination and Received: Revised: Accepted: Published: 11741

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Figure 1. Schematic illustration of experimental apparatus (tank diameter = 190 mm, liquid height = 190 mm), where α is the angle of inclination. In the figure, C is the impeller off-bottom clearance, T is the liquid height, e is the shaft eccentricity, and R is the tank radius.

the eccentric position of the inclined impeller on mixing time using a vessel with four baffles. From the aspect of mineral industries, however, not only the mixing time but also the power consumption are key factors for process design. Furthermore, baffles are frequently an obstacle for the reduction of power consumption and the enhancement of mass transfer for solids suspension.13,14 The objective of this work was, therefore, to introduce impeller inclination into a vessel without baffles to enhance the mixing quality by eliminating the segregated regions that are usually present above and below the impeller under laminar conditions. The motivation of this modification was to alter the symmetric flows inside the stirred vessel, aiming to attain complete mixing conditions. The mixing time and power input were as the key parameters in an attempt to determine the effects of the impeller off-bottom clearance and eccentricity on the mixing performance. In addition, two computational fluid dynamics (CFD) approaches are proposed to demonstrate the performance enhancement in both mixing of two miscible liquids and dispersion of fine particles using an inclinedimpeller mixing vessel. In general, the results are applicable to the overall goal of developing optimum industrial agitator designs to reduce both mixing time and energy input for

blending liquids in mixing tanks, where installation of baffles is not always necessary.

2. EXPERIMENTAL SECTION 2.1. Test Rig. Flow-pattern visualization experiments were carried out in a small mixing research rig (Figure 1) consisting of a 190-mm-diameter cylindrical tank with a flat bottom placed inside a rectangular outer acrylic tank. The outer tank was filled with tap water/glycerol to minimize the optical distortion. The shaft angle (α) was varied from 0° to 30° in our experiments. Sequential digital images were taken with a digital video camera to reveal isolated mixing region (IMR) structures and the flow pattern during the experiment. To capture the cross-sectional areas of the IMRs, a plane sheet of light passing through the 5mm slits at the center of a piece of cardboard located on both sides of the tank illuminated the IMRs, as shown in Figure 1. 2.2. Impeller. A six-bladed disk turbine (DT6) of diameter 70 mm was used in the tests to study the flow patterns for the Newtonian fluids. Photographs of the impeller used in this study are shown in Figure 2. The dimensions of the impeller are also shown in the figure. 2.3. Test Fluids and Experimental Methodology. Glycerol (99.7%) was employed as the working fluid, and the viscosity of glycerol was found to range from 0.80 to 0.90 Pa·s 11742

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Figure 2. Photographs of the DT6 impeller (with dimensions) used in this study: (a) front view, (b) top view.

at room temperature (22 °C). The neutralization reaction of NaOH and HCl was the basis for distinguishing different regions in the laminar Newtonian systems. A passive tracer fluorescent dye, as a pH indicator, was injected and homogenized in the working fluid. To remove all air bubbles, the working fluid mixed with fluorescent dye was allowed to settle for about 12 h before the experiments were conducted. The working fluid mixed with fluorescent dye was made basic by adding a basic solution consisting of 10 mL of 2 M NaOH. The working fluid was then stirred at 500 rpm for ∼2 h to fully disperse the base throughout the tank, leading to a uniform color distribution. After the agitator had reached the desired speed, a small amount of acidic solution (10 mL of 2 M HCl) was injected at the blade tip, and consequently, decolorization took place in the active mixing regions. In this study, the dimensionless mixing time (Ntm) was used to evaluate the overall mixing efficiency for different configurations. Note that N is the impeller speed, and tm is the decolorization time in the system. Ntm, on the other hand, also indicates the number of agitator rotations required to eliminate the IMRs in the laminar systems.

Figure 3. Meshes of (a) an unbaffled tank and (b) a Rushton turbine (impeller).

acceleration vector (m/s2). In our case, μe = 0 Pa·s, as the flow is in the laminar flow regime. Equations 1 and 2 were solved by the SIMPLE algorithm, assuming slip velocity at the liquid surface and nonslip velocity at the solid surface. The mesh models of the unbaffled tank and the impeller are shown in Figure 3.

4. RESULTS 4.1. Conducitivity Measurements. Figure 4 shows the progression of conductivity with time. The experiments were conducted by injecting the same amount of acidic solution (10 mL) into the active mixing region (AMR) and into the isolated mixing region (IMR) in the fresh working fluids. The locations of the IMRs and AMRs were determined by earlier experiments. The conductivity probe was placed on the liquid surface next to the sidewall (Figure 4). The difference in conductivities between the two experiments can be attributed mainly to limited interactions at the boundaries between IMRs and AMRs. It is hypothesized that diffusion is the only mechanism causing material exchange between IMRs and AMRs and that this dominant process is extremely inefficient in practical mixing processes. These conductivity tests suggest that a concentric system with installation of an upright impeller is infeasible in liquid blending, and we were thus motivated to explore alternative configurations to improve the mixing effectiveness.

3. COMPUTATIONAL MODEL The numerical simulations were performed using commercial CFD code RFLOW (Rflow Co., Ltd.). The details of our numerical approach to particle motion can also be found in ref 15. The equation of continuity is given by ∂ρ + (v·∇)ρ + ρ∇·v = 0 (1) ∂t and the equation of fluid motion is given by ⎡ ∂v ⎤ ρ⎢ + (v·∇)v⎥ = −∇p + ∇[(μ + μe )∇v] + ρ g ⎣ ∂t ⎦ (2) where v is the velocity vector (m/s), p is the pressure (Pa), ρ is the density of liquid (kg/m3), μ is the viscosity of fluid (Pa·s), μe is the eddy viscosity (Pa·s), and g is the gravitational 11743

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Figure 6. Velocity fields at different shaft angles: (a) 0°, (b) 30°. Re = 60.

Figure 4. Progression of conductivity versus time (red arrow, injection point at AMR; blue arrow, injection point at IMR) for the injection of 10 mL of 2 M NaOH. The probe was placed on the liquid surface and close to the tank wall as shown in the figure. Dashed lines indicate equilibrum values. DT6 impeller. Reynolds number (Re) = 60.

be improved by altering the shaft angle. It was found, however, that a small change in the shaft angle (10° of inclination) was not sufficient to avoid incomplete mixing conditions after a long agitation (e.g., 1 h), given that the green isolated zone remained segregated from the rest of the system, as shown in Figure 5b. It should be noted that the liquid temperature was maintained at room temperature and, thus, the viscosity was not significantly affected even after a long agitation.

4.2. Effects of Angle. In this section, the test results with shafts inclined at 10°, 20°, and 30° are presented. Photographs of the “quality” of mixing at different angles of inclination are shown in Figure 5. The Reynolds number was kept at 60. Notably, the absence of isolated mixing regions was found at α = 20° and α = 30°, implying that the level of homogeneity can

Figure 5. Effects of shaft angle on global mixing efficiency: (a) conventional design with a vertical shaft (α = 0°) and designs with shaft angles of α = (b) 10°, (c) 20°, and (d) 30°. Re = 60. DT6 impeller. 11744

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Figure 7. IMR centers at different shaft angles.

Figure 9. Effects of (a) impeller speed on specific power (W/m3) and mixing time (min), (b) Reynolds number on power number and dimensionless mixing time, and (c) impeller speed on specifc energy input (kJ/m3). Fluid viscosity = 0.85 Pa·s. The impeller was inclined at 20°. Figure 8. (a) Power consumption and mixing time at different shaft angles with the impeller maintained at the center position of the tank (i.e., C/T = 0.5 and e/R = 0, where C is the impeller off-bottom clearance, T is the liquid height, e is the shaft eccentricity, and R is the tank radius; see Figure 1). (b) Energy input into the system at different shaft angles. DT6 impeller. Re = 60.

and that a larger angle of inclination reduces the ratio of IMRs to AMRs. Typically, at α = 30°, we found that the ratio was reduced to 0. Figure 7 shows the shifts in the IMR position resulting from changes in the shaft angle, and this implies that the effects of the tank bottom and sidewall and the open fluid surface on the chaotic advection in chaotic regions (AMRs) are substantial. These effects are particularly significant at larger shaft angles (e.g., α = 20°, 30°), as the IMR centers move closer to the boundaries between regions. One can thus probably conclude that the reduction/destruction of IMRs in the angled-shaft systems is mainly due to the effect of breaking the spatial symmetry and shifting the locations of IMRs and AMRs. The effects of the shaft angle on power consumption are shown in Figure 8a. It is apparent that an increase in shaft angle

The destruction of IMRs in the angled-shaft tanks is likely due to the effect of breaking the spatial symmetry. For the upright impeller configuration, as expected, the velocity field was highly symmetric (see Figure 6a), whereas, as is evident in Figure 6b, the velocity field was dramatically altered in the angled-shaft system and the symmetry of the velocity was broken. It is thus easily conjectured that the breaking of the spatial symmetry introduces a greater degree of chaotic mixing 11745

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Figure 10. Experimental results from the inclined-shaft systems. (A−C) Effects of eccentricity (C/T = 0.50): e/R = (A) 0.58, (B) 0.26, and (C) −0.37. (D−F) Effects of clearance (e/R = 0): C/T = (D) 0.63, (E) 0.32, and (F) 0.11. (G−I) Effects of unconventional positions: (G) C/T = 0.24, e/R = 0.47; (H) C/T = 0.42, e/R = 0.26; and (I) C/T = 0.73, e/R = −0.37. Incomplete mixing in the presence of IMRs was noticed in cases C and D at Ntm = 10000, as indicated by a red cross symbol (×). A negative value for e/R means that the impeller was located on the right-hand side of the vertical center line. The shaft was inclined at 20°, and Re = 60.

0° and α = 10° are not shown in the figure, as they were essentially infinite, in that IMRs were still present below the impeller after a long agitation (∼1 h) and neither system achieved complete mixing (see Figure 5a). Figure 8b shows a comparison of the specific energies required to complete homogeneous mixing for different shaft angles. The specific energy (J/m3) calculated for complete mixing (here based on decolorization) was calculated for each system as

led to an increase in power consumption. Nevertheless, the 10% increase in power from 0° to 30° can be considered insignificant. The normalized mixing time (Ntm) is also included in the same figure. As can be seen in Figure 8a, the reduction (by ∼55%) in the mixing time due to the change in angle from 20° to 30° was substantial, implying that a larger shaft angle is favorable in blending liquids under laminar conditions, as the mixing time can be reduced dramatically through the removal of isolated mixing zones. It should be noted that, in these tests, the impeller was maintained at the center position of the tank and had a vertical shaft (i.e., C/T = 0.5 and e/R = 0, where C is the impeller offbottom clearance, T is the liquid height, e is the shaft eccentricity, and R is the tank radius). The mixing times for α =

E= 11746

P tm V

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Figure 11. Effects of (a) shaft eccentricity on power consumption and mixing time at C/T = 0.50 and (b) impeller clearance on power consumption and mixing time at e/R = 0. The impeller was inclined at 20°, and Re = 60. Figure 12. Effects of (a) shaft eccentricity on energy input at C/T = 0.50 and (b) impeller clearance on energy input at e/R = 0. The impeller was inclined at 20°, and Re = 60.

where P/V is the specific shaft power consumption (W/m3) and tm is the time (s) required to completely destroy the IMRs in the laminar systems, which is equivalent to the mixing time. Figure 8b shows the specific energy requirements for standard and modified configurations. For the 30° system, 420 kJ/m3 is required, whereas for the 20° system, 840 kJ/m3 is required, suggesting that the configuration with a larger angle is favorable in liquid blending because of its improved effectiveness in completing liquid homogenization. 4.3. Effects of Impeller Speed. The effects of impeller speed on the power consumption and mixing time are shown in Figure 9a. Power consumption increased and mixing time decreased with increasing speed, as expected. Figure 9b shows that the power number and dimensionless mixing time decreased with increasing Reynolds number, as reported in the literature (see, for example, the article by Takahashi et al.,12 although their tank was installed with baffles). The comparison between Takahashi et al.’s data12 and our data in Figure 9b suggests that, at higher Reynolds numbers (Re > 60), the effect of baffles on the mixing time is almost negligible in angled-shaft systems. Figure 9c shows that the specific energy required for complete mixing decreased with increasing impeller speed. For a given viscosity, this suggests that it is more energyefficient to operate at higher impeller speed for the purpose of complete homogenization.

4.4. Effects of Impeller Clearance and Eccentricity. The test cases for various offset impeller configurations used to determine the effects of impeller clearance and eccentricity are illustrated in Figure 10. In all of these cases, the impeller shaft was inclined at an angle of 20°, and the impeller speed was kept at 500 rpm, giving Re = 60 (i.e., laminar flow conditions). At Re = 60, it was observed that all of the test configurations shown in Figure 10 provided complete mixing (i.e., removal of IMRs), except for cases C and D, as indicated in the figure. Case C had the impeller located on the right-hand side of the vertical center line (e/R = −0.37), and case D had the impeller located above the horizontal center line (C/T = 0.63). e is defined as the eccentric distance (see Figure 1), and R is the tank radius, so an e/R value with a positive sign means that the impeller is located on the left-hand side of the vertical center line, whereas a negative sign indicates that the impeller is located on the right-hand side (see Figure 10). Figure 11a shows the power consumption and dimensionless mixing time (Ntm) for the DT6 impeller at different eccentric locations in the stirred tank with constant impeller off-bottom clearance (C/T = 0.50). The data suggest that a larger eccentricity results in a slight increase in power consumption and a dramatic reduction in mixing time. 11747

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Figure 13. CFD results: Mixing of two miscible liquids in a conventional agitation system (α = 0°) for Re = 60 as a function of time: t = (a) 0, (b) 0.5, (c) 1, (d) 1.5, (e) 2, (f) 3, (g) 4, and (h) 5 s.

interest to consider the effects of this finding for real-world applications. Two cases based on CFD are illustrated here to show the benefits of using angled-shaft systems. 5.1. Mixing of Two Miscible Liquids. Figures 13 and 14 show the results of mixing of two miscible liquids in two configurations, namely, at α = 0° and α = 30°, respectively. Note that the liquids (90% blue liquid in the lower part of the vessel and 10% red liquid in the upper part of the vessle) used in the simulations had identical viscosities and densities. The immediate observation from Figure 13 is that highly inhomogeneous mixing conditions are present inside the standard upright-shaft system. Much faster mixing can be observed in the upper region of the vessel as compared to the lower region. Although this is most likely due to the initial distribution of liquid layers (see Figures 13a and 14a), the highly inhomogeneous situation was not found in the angledshaft tank, as shown in Figure 14. Based on the results in Figures 13 and 14, we hypothesize that the material exchange between the lower region of the tank (below the impeller) and the upper region (above the impeller) is very slow for conventional upright-shaft systems, whereas for angled-shaft systems, the breaking of the spatial symmetry can dramatically enhance the mixing between these two regions. 5.2. Dispersion of Solid Particles. Disperisons of fine paritcles are commonly encountered in the process industry. Thus, it is also of interest to examine the effects of shaft angle on the solid−liquid mixing peformance in agitation systems. The numerical apporach was based on the following

Figure 11b shows the effects on the power and the dimensionless mixing time of varying the impeller-to-bottom clearance at e/R = 0. It can be seen that that the impeller-tobottom clearance had a neligble effect on the specific power, whereas an increase in impeller clearance led to a substantial increase in mixing time. This figure implies that a lower impeller clearance is preferred for liquid blending, considering that the mixing time can be reduced in comparsion to that required for higher impeller clearances and the power consumption can be maintained. Figure 12 shows the specific energy input required to achieve complete mixing decolorization for different systems corresponding to those showing in Figure 5. It can be seen from Figure 12a that increasing the eccentricity from 0 caused a reduction in the energy input, whereas the effect of impeller eccentricity was negligible in the range of e/R = 0.26−0.73. Note that the data for negative e/R values are not included here, because IMRs were still present in the system, as shown in Figures 10C and 10D. Figure 12b shows that an increase in C/ T led to a substantial increase in the specific energy input. Based on Figure 12, it is hypothesized that case G in Figure 10 is the most energy-efficient setup, as it has largest e/R value and the smallest C/T value.

5. CFD SIMULATIONS The experimental measurements discussed in the previous section indicated that a large angle of inclination can enhance mixing by removing isolated mixing regions. It will be of 11748

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Figure 14. CFD results: Mixing of two miscible liquids in an angled-shaft system (α = 30°) for Re = 60 as a function of time: t = (a) 0, (b) 0.5, (c) 1, (d) 1.5, (e) 2, (f) 3, (g) 4, and (h) 5 s.

measured in terms of the specific energy required for complete mixing (i.e., full homogenization), less energy is required at increased shaft angles. It is interesting to note that, in general, shortened mixing times can be achieved at increased speeds with the penalty of increased power requirements, as expected. An obvious question is: Is it better to mix rapidly (i.e., short tm) at high speeds and large power input or to mix slowly (i.e., long tm) at low speeds and less power input? The answer here is that it is actually more energy efficient to operate at high speeds considering that less specific energy input can be attained in these cases. It should be noted that increased speeds in largescale industrial mixing vessels have engineering limitations, such as high erosion wear on the impeller blades, shaft vibrations, and so on. In those cases, operating the system at high speed might not be acceptable. It was also demonstrated in this study that further optimization can be achieved by altering the shaft eccentricity and the impeller clearance. Nevertheless, in the optimization process for the large-scale mixing tanks used in the minerals processing industry, the use of an off-center inclined impeller should be carefully considered, as it might produce some stagnant mixing regions, particularly in areas that are far from the impeller. The concept of the specific energy required for complete mixing homogenization is a more reasonable approach, as traditionally, the mixing time and the power are quantified

assumptions: Particles of the same density as the liquid (ρ = 1261 kg/m3) were used, and they were distributed evenly just above the liquid level at t = 0 s (see Figures 15a and 16a). The solids concentration was 1% (v/v) for both the standard and angled-shaft systems. Figure 15 shows the distribution of particles as a funciton of time in the standard mixing vessel. The performance of this system was rather poor, considering that most of the flowing particles were restricted to the upper region of the vessel and only a few particles were able to travel to the lower part of the vessel. Such an inhomogeneous distribution will certainly affect product quality and eventually lead to the formation of large amounts of undesirable byproducts. Therefore, the conventional mixing system is not favorable for handling fine particles. Notably, the inclined-shaft configuration promoted much better particle dispersion, as shown in Figure 16. It is very interesting to note that these results are analogous to those obtained for the mixing of two misciable liquids, as shown in Figure 14. The CFD results confirm that homogeneous conditions can be achieved in both liquid−liquid and liquid− soild systems by using inclined-impeller systems.

6. DISCUSSION The results presented in this article suggest that an agitator with an inclined shaft could be beneficial because it can provide better mixing through the destruction of isolated mixing regions (IMRs) for a modest increase in the power consumption. When 11749

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Figure 15. CFD results: Dispersion of fine particles in a conventional system (α = 0°) for Re = 60 as a function of time: t = (a) 0, (b) 0.5, (c) 1, (d) 1.5, (e) 2, (f) 3, (g) 4, and (h) 5 s.

Figure 16. CFD results: Dispersion of fine particles in an angled-shaft system (α = 30°) for Re = 60 as a function of time: t = (a) 0, (b) 0.5, (c) 1, (d) 1.5, (e) 2, (f) 3, (g) 4, and (h) 5 s.

the mixing time scale is finite. In cases where IMRs are present, the mixing time becomes very long, eventually leading to infinite specific energy requirements; therefore, mixing time

separately. This approach is obviously an advantage in that it allows for the comparison of different designs in terms of the energy required. This approach is applicable, however, only if 11750

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cannot be used for design optimization. The direct comparison of shaft power is more relevant for such cases (e.g., with IMRs). The CFD results presented in this article also confirmed that inclined-shaft configurations have additional advantages in comparison to conventional setups. It is strongly suggested from these results that the simple method of employing inclined-shaft configurations can be used to potentially reduce reaction byproducts and improve chemical reaction rates by promoting better mixing homogeneity. However, further experiments are strongly recommended to verify the accuracy of our CFD results. Finally, it is useful to mention that the current results were obtained with baffles removed. This is of practical significance, as baffles are sometimes difficult to clean and descale in some industrial applications. It is therefore of interest to study the energy efficiency without baffles installed when optimizing the impeller installation configuration, that is, the shaft angle or offset location. It should be mentioned that, with baffles installed, the energy efficiency generally improves; see our separate work on this effect in ref 7.

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7. CONCLUSIONS In this study, mixing was characterized through experiments by means of decolorization visualization. The experimental results revealed that systems with shafts inclined by ≥20° can dramatically enhance the mixing quality and reduce the mixing time. At a greater shaft angle, more power input is required but for a shorter mixing time; larger angles also lead to lower energy requirements. The results also suggest that, for an agitator with an angled shaft, it is more energy efficient to operate at high impeller speed. For systems with an inclined shaft, at constant off-bottom clearance (e.g., C/T = 0.5), an increase in impeller eccentricity leads to a reduction in power but results in an insignificant increase in mixing time. At constant eccentricity (e/R), the impeller off-bottom clearance does not affect mixing time, whereas an increase in off-bottom clearance gives rise to an increase in power consumption. The tradeoff between power input and mixing time should be fully considered in the design of a tank with an inclined shaft. Our results lead to the conclusion that the energy input into a mixing system can be minimized by altering e/R and C/T. A setup with large e/R and C/T values is considered to be the optimum system based on this study. The CFD results confirmed that both liquid blending and particle dispersion can be dramatically enhanced by using an inclined shaft in the mixing system.

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D = impeller diameter (m) e = shaft eccentricity (m) g = gravitational acceleration vector (m/s2) N = impeller speed (rev/s, rpm) Np = power number Ntm = dimensionless mixing time P = power (W) p = pressure (Pa) R = tank radius (m) T = liquid height (m) tm = complete decolorization time (min) V = tank volume (m3) v = velocity vector (m/s) α = angle of inclination (deg) μ = fluid viscosity (Pa·s) μe = eddy viscosity (Pa·s) ρ = liquid density (kg/m3)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +61 3 9545 8380. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors greatly appreciate the support of Dr. Degang Rong, Rflow Co., Ltd., Soka, Saitama, Japan. S.W. was funded by the CSIRO Fluids Engineering group and Monash University.

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NOMENCLATURE C = impeller off-bottom clearance (m) 11751

dx.doi.org/10.1021/ie401003s | Ind. Eng. Chem. Res. 2013, 52, 11741−11751