Ind. Eng. Chem. Res. 2009, 48, 1665–1672
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Turbulent and Laminar Mixings in an Unbaffled Agitated Vessel with an Unsteadily Angularly Oscillating Impeller Masanori Yoshida,*,† Yasuyuki Nagai,† Kazuaki Yamagiwa,† Akira Ohkawa,† and Shuichi Tezura‡ Department of Chemistry and Chemical Engineering, Niigata UniVersity, Niigata 950-2181, Japan, and Shimazaki Mixing Equipment Co., Ltd., Tokyo 116-0014, Japan
For an unbaffled agitated vessel with an angularly oscillating impeller whose rotation continues while periodically reversing its direction at a set angle, the progress of mixing of Newtonian liquid phase was studied experimentally in relation to the impeller power characteristics. Results obtained for the angular oscillation mode of a disk turbine impeller with six flat blades with different amplitudes (Strouhal number) were compared to those obtained for the unidirectional rotation mode of the impeller of identical design. Using an angularly oscillating impeller, turbulent and laminar mixings were effective, respectively, with larger and smaller amplitudes. The mixing characteristics for turbulent and laminar flow regions were discussed on the basis of the flow field features, which the impeller power number reflected. 1. Introduction A vessel agitated using a mechanically rotating impeller is widely used as a reactor in chemical processes. The performance of agitation apparatuses depends strongly on the working liquid viscosity. Consequently, due to variation of the liquid viscosity, modification of the impeller design or change of its geometrical conditions is conventionally alternative for effective operation. Mixing operations by which the viscosity increases because of chemical and physical changes of liquid phase are all too common in process productions. The viscosity of liquids involving products is often larger than that of liquids involving reactants, sometimes by several orders of magnitude. Such systems create challenging problems in industrial practice. Indeed, it is seldom possible to modify the impeller design or change its geometrical conditions with progress of the processing step. When the liquid viscosity builds up, two mixing mechanisms are usually required: an intensive dispersion at low viscosity and a sufficient homogenization at higher viscosity. Unfortunately, no existing agitator meets those requirements. Therefore, one must resort to an apparatus that represents the best possible compromise in terms of performance over the range of viscosities observed in the process. A concept proposed in the literature for the requirements described above is the combination of different impeller.1-3 In some cases, the respective impellers are rotated at different speeds using independent motors.4-8 Treatment of the problem merely by changing the easily controlled operating conditions can simplify the apparatus configuration. For a previous study, we developed an agitator with an impeller forward-reverse rotating over a variable angle range, angular oscillation.9 This system, which uses a pulse motor, enables setting of the amplitude and frequency of the impeller angular oscillation as operating conditions, and thereby conveniently supplies angular oscillations. In an unbaffled vessel with such an agitator installed, a flat liquid surface is formed even when low-viscosity liquids are agitated. A growing interest has arisen in such an agitation mode, in which an impeller produces unsteady fluid flows within a vessel by periodically changing either the * To whom correspondence should be addressed. Tel.: +81-25-2627343. Fax: +81-25-262-7343. E-mail:
[email protected]. † Niigata University. ‡ Shimazaki Mixing Equipment Co., Ltd.
direction or the speed of rotation, that is, unsteady rotation.10-16 Few prior studies for vessels agitated by an unsteadily rotating impeller address processing for change of the liquid viscosity ranging from a turbulent flow region to a laminar one. For vessels of this type, investigation of mixing characteristics with variation of the liquid viscosity would provide an operational guideline in the systems of different viscosity and evaluate applicability to systems of variable viscosity. In this work, the liquid phase mixing in the unbaffled vessel agitated by an angularly oscillating impeller was investigated experimentally for Newtonian liquids with lower and higher viscosities. The impeller power characteristics relating to the mixing characteristics of the vessel were first evaluated. The mixedness in the mixing process was then expressed in terms of the intensity of segregation of the bulk liquid. Furthermore, the mixing characteristics in the turbulent and laminar flow regions were compared to those for the vessel agitated by a unidirectionally rotating impeller. 2. Experimental Section A schematic diagram of the experimental setup is presented in Figure 1. An unbaffled vessel with a flat base made of transparent acrylic resin (300 or 150 mm inner diameter, Dt) was used. Regarding respective sizes of vessels, the liquid depth was held at Dt. Disk turbine impellers with six flat blades (0.25Dt or 0.50Dt diameter, Di) were used. The impeller was set at a height of 0.50Dt from the bottom. The pulse motor driving the shaft provides sinusoidal angular oscillations with different amplitudes and frequencies. When the angular displacement is expressed in the form of a cosine function, the impeller rotation rate, Nr, is given by a sine function as Nr ) θoNfr sin(2πNfrt)
(1)
where θo is the amplitude of impeller angular oscillation, and Nfr is its frequency. The average impeller rotation rate, Nrav, was defined as the average over a period of the absolute Nr values. According to eq 1, Nrav is related to θo and Nfr, which are the operating conditions, as Nrav ) (2/π)θoNfr
(2)
Therein, θo and Nfr were set for Nrav to (100/60)-(200/60) s-1. Control experiments using the identical design of the
10.1021/ie801502q CCC: $40.75 2009 American Chemical Society Published on Web 12/24/2008
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Figure 1. Schematic diagram of the experimental apparatus (dimensions in mm).
impeller in unidirectional rotation mode were carried out at Nr of (100/60)-(200/60) s-1, where lower-viscosity liquids were agitated in the vessel fitted equiangularly with four baffles of 0.1Dt width and higher-viscosity liquids were agitated in the unbaffled vessel. The impeller power consumption, Pm, under a set condition of the rotation rate, Nr, was determined by measuring the torque, Tm, with strain gauges fitted on the shaft, as follows. Pm ) 2πNrTm
(3)
Although Pm in unidirectional rotation mode of the impeller is time-independent, that in angular oscillation mode of the impeller is time-dependent because of periodic changes of Nr and Tm. For the latter mode, the average weighted for the time, Pmav, was calculated from the following equation based on the total energy transmitted over one cycle:17 Pmav )
∑P
m∆t/(1/Nfr)
(4)
The bulk liquid flow was visualized using the particle suspension method, and the flow pattern was characterized. Details of the method were described in a previous report.18 Batch experiments on the liquid phase mixing were performed under conditions of Dt ) 300 mm, Di/Dt ) 0.25 for a system mixing a lower-viscosity liquid, and under conditions of Dt ) 150 mm, Di/Dt ) 0.50 for a system mixing a higher-viscosity liquid. Investigation of the lower-viscosity liquid (water) was based on the concentration fluctuation of the electrolytic tracer with progress of dilution after its release in concentrated amounts into the bulk.19 An electrolyte (sodium chloride) solution was prepared with the concentration of 1 mol/L. One twentythousandth of the volume of that liquid was fed rapidly from the upside of the impeller rotating at the determined rate and the neighborhood of the shaft through a magnetic valve connected to a limit switch (Figure 2a). The sodium chloride concentration of the liquid phase was determined by measuring the electrical conductivity with electrode cells made of platinum wire. Investigation of higher-viscosity liquid was based on the
Figure 2. Schematic flow diagram of mixing experiments (dimensions in mm). (a) System mixing lower-viscosity liquid. (b) System mixing higherviscosity liquid.
concentration fluctuation of the iodine with progress of its reduction reaction by sodium thiosulfate in the liquid phase.20The iodine plus potassium iodide and the sodium thiosulfate were dissolved respectively in Newtonian millet jelly solutions, and their viscosity was adjusted to the set value. The concentration of iodine was 0.4 × 10-3 mol/L, and that of sodium thiosulfate was 1.2 × 10-3 mol/L, corresponding to 1.4 times the amount of the equivalent for the reaction. The sodium thiosulfate solution was fed to a height of 0.5Dt, and over it, the iodine solution was decanted to a height of Dt.21Subsequently, the impeller rotation was started at the determined rate (Figure 2b). The concentration of iodine in the liquid phase was determined by measuring the tone of the image of the bulk liquid, which was taken under a set condition of illuminance, on a gray scale of 1-256. In the lower-viscosity system, measurements were made at 36 points on the vertical plane, including the shaft, for the four test planes varied in the circumferential direction. That is, the liquid part within the vessel was divided into 144 compartments with volume Vi, as depicted in Figure 3a. For the higher-viscosity system, an assumption was made that a side-view image reflected states in the region including the depth. Under this assumption, the 36 compartments with volume Vi, as depicted in Figure 3b, were configured, and the results obtained for the four aspects were averaged. The intensity of segregation of the bulk liquid in the mixing process, S, is expressed in terms of the deviation of concentration, Ci, as follows.22 S)
[∑ {V (C -C*)} ⁄ ∑ {V (C -C*)} ]
2 1⁄2
2
i
i
i
o
(5)
In that equation, Co and C* respectively signify the follower concentration in the bulk liquid at the start of mixing and that when mixing is completed.
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Figure 4. Relationship between Np and Rem.
Therein, F is the liquid density. Regarding the angularly oscillating impeller, application of the average power consumption, Pmav, to Pm in eq 6 and substitution of the average rotation rate, Nrav, for Nr in eq 6 yields the equation presented below. Np ) (π/2)3Pmav /F(θoNfr)3D5i
(7)
Although on Np in the unidirectional rotation mode of impeller the effect of liquid flow produced by the impeller is described with the Reynolds number, the time dependence of the impeller rotation must be considered for analysis in the angular oscillation mode of impeller. Dimensionless analysis for the latter mode provides the following correlation form.23 Np ) func.(Rem,Stm)
(8)
Rem is the impeller Reynolds number and is expressed as Rem ) F(NrDi)Di/µ ) FNrD2i /µ
(9)
where µ is the liquid viscosity. In fact, Stm represents the Strouhal number, which is related to the ratio of the local acceleration of the flow to its convective acceleration, and then that for the angularly oscillating impeller is reduced to the ratio of the frequency of oscillation, Nfr, to Nr. Stm ) NfrDi/(NrDi) ) Nfr /Nr
(10)
Using eq 2 for eqs 9 and 10, the relations between the respective parameters and the operating conditions and physical properties are obtained.
Figure 3. Test compartment for mixing examination (dimensions in mm). (a) System mixing lower-viscosity liquid. Baffles in unidirectional rotation mode were arranged on the cross sections 1 and 3. (b) System mixing higherviscosity liquid.
3. Results and Discussion 3.1. Power Consumption of Impeller. Before the mixing process was analyzed, the power characteristics reflecting the agitation action through the impeller were examined. Power consumption of the unidirectionally rotating impeller, Pm, is made dimensionless as the power number, Np, the characteristic length being the impeller diameter, Di, and the characteristic velocity being the product of Di and the impeller rotation rate, Nr. Np ) Pm/F(NrDi)3D2i ) Pm/FNr3D5i
(6)
Rem ) (2/π)F(θoNfr)D2i /µ
(11)
Stm ) (π/2)/θo
(12)
From the viewpoint of operating conditions, variations of θoNfr and θo, respectively, correspond to changes of Rem and Stm. Actually, Np, as determined experimentally, was investigated as the function of Rem and Stm calculated from eqs 11 and 12. Next, the effects of operating conditions and physical properties on Np were evaluated. Figure 4 shows the relation between the power number, Np, and the impeller Reynolds number, Rem, with the impeller Strouhal number, Stm, as a parameter. For any Stm, Np was almost inversely proportional to Rem for Rem smaller than 1. The rate of decrease of Np lessened with increased Rem, and Np tended to be practically unchanged in the Rem range larger than 1000. These results are considered to demonstrate that the drag force acting on the impeller blades conformed to Newton’s law for the turbulent flow in the higher Rem range24 and conformed to Stokes’ law for the laminar flow in the lower Rem range.25
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Figure 5. Flow patterns produced by impeller at Rem of 5200.
Figure 7. Turbulent mixing process as viewed from change in segregation intensity.
Figure 6. Time-courses of sodium chloride concentration in liquid phase (compartment number is given in Figure 3).
For the effect of Stm, Np was found not to differ significantly depending on Stm throughout the range of Rem. The results for the unidirectional rotation mode are additionally presented in Figure 4. The results reported in the previous works26,27 were added to the figure. For Rem larger than 1000, the values of Np for the unbaffled vessel with the angularly oscillating impeller were about 1.5 times those for the baffled vessel with the unidirectionally rotating impeller. This difference seems to be mainly attributable to the existence of the disturbed flow region behind the blades, which is a factor relating to the magnitude of the resistance of liquid against the impeller motion, the form drag on the
impeller, extended because of the unsteadiness of the impeller motion.28 For Rem of smaller than 1, Np exhibited a nearly identical value irrespective of the agitation mode. Additionally, for the unidirectional rotation mode, Np is independent of the vessel geometry with or without baffles. This result confirms that the viscous drag on the impeller plays the dominant role in the fluid resistance. Accordingly, weak centrifugal action with the impeller rotation can be predicted for such a low Rem range.29 3.2. Turbulent Mixing. The liquid flow produced by the agitating impeller acts on fluids within the vessel to enhance phenomena such as liquid-phase mixing. Figure 5 depicts images showing the liquid flow on the vertical plane, including the shaft within the vessel for the Reynolds number, Rem, of 5200. That recording was made with exposure for the time equal to the rotation cycle, which was long enough to provide an image outlining the flow pattern. The flow pattern for the angularly oscillating turbine type impeller, independent of the amplitude, θo, resembled that for the unidirectionally rotating turbine: a well-known pattern with a characteristic radial discharge stream and two circulation loops. According to such a flow pattern, when the tracer liquid is released as described above (Figure 2a), the tracer mass is considered for any agitation mode to be
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Figure 8. Flow patterns produced by impeller at Rem of 1.1.
discharged and circulated after deformation by shearing action through the impeller. Figure 6 shows typical time-courses of the sodium chloride concentration in the liquid phase after release of the tracer liquid. The pattern of the time-course, the process for the concentration to approach the steady value, is divided into two broad categories:30 a pattern with decrease following overshoot shortly after release of the tracer liquid, and monotonic increase. For any agitation mode, the latter pattern was generally observed in regions distant from the impeller. The former pattern tended to come in the impeller discharge region. Especially for the angular oscillation mode with θo of 45°, the tendency was notable. Because of such difference of the mixing pattern between the circumferential positions, that difference is considered to be attributed to the difference of circumferential convection depending on θo. For the angular oscillation mode with θo of 45°, although insufficient mixing existed in the circumferential direction, the distribution of concentration was rather uniform in the other directions. This result demonstrates that the angular oscillation of the impeller, which provides the turbulent flow
extending over its surrounding region, is effective for overall mixing throughout the vessel. Figure 7 shows the nonuniformity of the concentration distribution within the vessel in the mixing process in terms of the segregation intensity of the bulk liquid, S. Irrespective of the agitation mode, mixing proceeded exponentially and completely, with S becoming almost 0 after an elapsed time. Although the difference of S depending on the agitation mode was indistinct in an early stage of the mixing process, S in its middle stage was lower for the angular oscillation mode with the amplitude, θo, of 135°, which demonstrated a larger effect of agitation by the angularly oscillating impeller with a larger amplitude on mixing enhancement. Mixing progress with a lower segregation intensity can be interpreted as a reflection of the action of liquid flow provided by angular oscillation of the impeller. That is, the liquid mass is deformed effectively, and the divided elements are circulated and thereby distributed uniformly throughout the vessel. 3.3. Laminar Mixing. Images outlining the liquid flow pattern within the vessel for the Reynolds number, Rem, of 1.1
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Figure 11. Laminar mixing process as viewed from change in segregation intensity.
Figure 9. Time-courses of iodine concentration in liquid phase (compartment number is given in Figure 3).
are depicted in Figure 8. The exposure time for recording the image was varied relative to the rotation cycle, which allowed us to compare the period necessary for the pattern to be detected. In angular oscillation mode, the two circulation loops were found for the flow pattern during several oscillations of the impeller, independent of the amplitude, θo. On the other hand, the intermissive loops were traced through observation for a few minutes in unidirectional rotation mode. That is, the inertia force of impeller angular oscillation is anticipated to contribute effectively to transmission of the motion. This comparative result suggests that the viscosity affects the angularly oscillating impeller lower than the unidirectionally rotating impeller. In the run examining the laminar mixing, as described above (Figure 2b), the boundary plane between the upper liquid layer of the iodine and the lower one of the sodium thiosulfate comes to be agitated in an early stage, where deformation by shearing action through the impeller is considered to govern the mixing progress.
Figure 10. Mixing states attained by different impeller agitations.
Figure 9 shows typical time-courses of the iodine concentration in the upper liquid layer after the start of the impeller rotation. A tendency was observed for all agitation modes: the decrease of concentration was fast in the region lying on the pathway of main flow, the circulation loop, and was slow in the region surrounding the center of the loop. For the difference between the regions, results obtained in angular oscillation mode revealed that the concentration distribution was more uniform at the amplitude, θo, of 45° and that the uniformity decreased concomitantly with the increase of θo. In unidirectional rotation mode, the difference of concentration depending on the position within the vessel, an isolated mixing region, remained even after long agitation.31,32 Figure 10 compares the state within the vessel after long agitation among the agitation modes. The isolated mixing region for the unidirectional rotation mode is known to be one like a doughnut. No observation of the relevant region for the angular oscillation mode seems to be caused by an alternating circumferential flow produced by the impeller. That is, because the circumferential flow reverses periodically its direction, no stable region may be formed. Relatively enhanced mixing with a smaller amplitude could be attributed to repetition of the alternation due to a larger frequency. This result demonstrates the angular oscillation of the impeller, which provides the unique shear deformation to the liquid mass within the vessel, is effective for the mixing in the flow field where a weak centrifugal action with the impeller rotation is expected. Figure 11shows the segregation intensity of the bulk liquid, S, in the mixing process. Similarly to the turbulent mixing, the
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difference of S depending the agitation mode was indistinct in the early stages of the mixing process. Actually, S in the middle stage or later exhibited lower values in order of the angular oscillation mode with the amplitude, θo, of 45°, that mode with θo of 135°, and the unidirectional rotation mode, which demonstrated a larger effect of agitation on the mixing enhancement by the angularly oscillating impeller with a smaller amplitude. According to this result, decreased amplitude, an increased local inertia as described by the Strouhal number, is useful for improving mixing in a flow field with ineffective inertias. 4. Conclusions For an unbaffled vessel agitated by an angularly oscillating disk turbine impeller with six flat blades, the progress of mixing of Newtonian liquid phase was weighed experimentally against that for a vessel agitated using a unidirectionally rotating impeller of identical design, considering the impeller power characteristics. The impeller power number was expressed as a function of the Reynolds number, irrespective of the Strouhal number. In the turbulent flow region, the power number of the angular oscillating impeller was larger than that of the unidirectionally rotating impeller. Contrastingly, the impeller power number in the laminar flow region was recognized to be independent of the agitation mode. With the bulk flow being turbulent, the disturbed flow region extended through the angularly oscillating impeller, whose higher power number reflected, was effective for overall mixing throughout the vessel. When operation was performed with a larger amplitude, mixing was confirmed to proceed with a lower segregation intensity, a higher mixedness. The deformation by shearing action with an increased local inertia through the angular oscillating impeller was effective for mixing in the laminar flow field where a weak centrifugal action with the impeller rotation can be expected because of the impeller power number independent of the agitation mode. Operation with a smaller amplitude achieved mixing progress with a higher mixedness. Nomenclature Ci ) NaCl or I2 concentration measured at different positions within the vessel (mol/L) Co ) NaCl or I2 concentration in bulk liquid at startup (mol/L) C* ) NaCl or I2 concentration in bulk liquid after full mixing (mol/L) Di ) impeller diameter (mm) Dt ) vessel diameter (mm) Nfr ) frequency of the angular oscillation of impeller (Hz) Np ) impeller power number (-) Nr ) rotation rate of impeller (s-1) Nrav ) average rotation rate of impeller (s-1) Pm ) power consumption of impeller (W) Pmav ) average power consumption of impeller (W) Rem ) impeller Reynolds number (-) S ) intensity of segregation (-) Stm ) impeller Strouhal number (-) t ) time (s or min) Tm ) torque of shaft with impeller (N · m) Vi ) volume of liquid within the compartment defined in Figure 3 (m3) Greek Symbols µ ) viscosity of liquid (Pa · s) θo ) amplitude of the angular oscillation of impeller (rad or deg) F ) density of liquid (kg/m3)
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ReceiVed for reView October 5, 2008 ReVised manuscript receiVed November 12, 2008 Accepted November 12, 2008 IE801502Q
