here, N is the number of sampling data. Ic and Ie are the image intensity of calibration curve and experimental data, respectively. The deviations of the image intensity, e at point_1 to point_9 are 0.0169, 0.0228, 0.0175, 0.0224, 0.0159, 0.0243, 0.0317, 0.0084 and 0.0106 respectively. As shown in Figure 4(c), for each sample point, one can find a considerable deviation as much as 30% in low concentration. The error is caused by the reflection and refraction of the source light on the cylindrical surface of the tank. Although the concentration rate is hard to be evaluated with a good accuracy at low concentration rates, we can discuss the mixing process qualitatively in terms of the change of concentration rate.
Figure 4. (a) Dyed water; (b) Image intensity of the dyed water; (c) Local calibration curve of fixed points (positions of points are described in Figure 3).
3. Results and Discussion
3.1. Visualization Results
The flow pattern of fin-type and conduit-type mixers have been compared in Figure 5 and Figure 6. By observation of motion pictures, the flow direction was determined by the motion picture, then the red arrows indicate the stream schematically.
(a) (b) (c)
Figure 5. Flow visualization of fin-type mixer at n = 350 rpm. (a) h = 3/4H, t = 150 s; (b) h = 1/4H, t = 50 s; (c) h = 3/4H, t = 15 min.
(a) (b) (c)
Figure 6. Flow visualization of conduit-type mixer at n = 350 rpm. (a) h = 3/4H, t = 150 s; (b) h = 1/4H, t = 40 s; (c) h = 3/4H, t = 15 min.
As shown in Figure 5(a) and Figure 5(b), the fin-type mixer generates a suction flow like tornado from the bottom, then, the flow turns towards the radial direction. In the condition in Figure 5(b), the particles are transported from the bottom and scattered to all direction. Also, it was observed that a small fraction of the particles near the bottom was moved towards the corner of the tank. Figure 5(c) shows the suspended tracer pattern after long duration time, t = 15 min. One can observe a lot of sedimentation on the bottom corner. It was confirmed that the fin-type mixer has poor performance about the stirring at the bottom corner. Though the stirring process of the fin-type mixer near the mixer is strong, the process degenerated at the long distance.
According to Figure 6(a) and Figure 6(b), a strong helical suction was induced by the conduit-type mixer. In this case, spiral flow was generated along a horizontal plane due to the centrifugal force. Near the tank wall, particle trajectory was divided into upward and downward direction. Figure 6(b), shows that particles can be reached up to the center from the corner of the tank.
After the long-elapsed time, Figure 6(c) shows a small amount of sedimentation at the bottom corner. This fact means that the agitation at the corner of the bottom of the tank by the conduit-type mixer is stronger than that in the fin-type mixer, as shown in Figure 5(c).
Additionally, it was observed that the central vortex and the wave at the upper surface by the fin-type mixer is larger due to the suction flow from the upper surface.
A comparison of flow patterns between the fin-type and the conduit-type based on the flow visualization experiment is illustrated schematically in Figure 7. The fin-type mixer emanates the fluid to the all radial direction. The kinetic motion of emerged fluid was dissipated in short distance. Then we can observe a couple of circulation combining with the two-sided suction, one from the upper region and another one from the lower region, thus, the fin-type mixer generates a closed loop around the mixer.
Figure 7. Schematic diagram of the flow pattern of (a) Fin-type mixer; (b) Conduit-type mixer.
On the other hand, during the rotation of mixer, the nozzles of the conduit-type mixer eject a strong radial jet due to the centrifugal force. Along the horizontal plane, the jets propagated toward the tank wall. On the tank wall, the jets was divided upward and downward along the side wall, which pushes the fluid towards the center from the corner of the tank. From the center of the tank bottom, a strong helical suction pulls the fluid into the mixer and ejects them again as a jet flow. The conduit-type mixer generates a wider and stronger circulatory flow pattern at the lower region of the mixer. Also, the circulation flow in upper region was constructed remarkably (see Figure 6(a) again) but weaker than the fin-type mixer because the central vortex and the wave at the upper surface was weak.
3.2. Concentration Measurement
The concentration measurement was carried out to investigate the time sequential progression of local concentration. Here, the characteristic mixing time should be determined relative to the equilibrium concentration rate. Mathematically, the mixing time tm can be estimated to satisfy the following condition as the following,
for whole field (3)
here, ci(t) is the concentration at time t, and t∞ is the time reaching up to the equilibrium state, and ε = 0.05.
As the sampling proves, 9 equidistant points, were chosen as described in Figure 3 along the vertical line with a radial distance of D/4. Figure 8 presents the results of the concentration experiment. The x-axis represents time in seconds and y-axis concentration rate. At the initial state, the concentration at the point_5 to point_9 should be 0%, however, some data included an error. This was caused by the reflection and refraction of the source light on the cylindrical surface of the tank as described in Section 2.2.
Figure 8. Concentration rate of points of interest with time (positions are described in Figure 3). (a) Fin type; (b) Conduit type.
The temporal variation of concentration at the 9 points from t = 0 to 720 s for fin-type is presented in Figure 8(a). In this figure, the concentration rates at point_1 to point_4 are decreasing after t = 70 s and those at the point_5 to point_9 increasing at t = 60, 130, 210, 280, and 330 s, respectively. The concentration rates for the latter points reach the maximum 80%, 68%, 57%, 53% and 52% at t = 160, 220, 270, 350, and 380 s, respectively. All points reached at equilibrium concentration till, tm = 440 s.
Figure 8(b) presents the concentration at the specific points by the conduit-type mixer. According to this figure, the concentration at point_1 to 4 is decreasing after t = 60 s and those at point_5 to 9 is increasing at a time approximately, t = 60, 180, 250, 320, and 390 s, respectively. The concentration rates at latter points reached maximum 78%, 70%, 60%, 55% and 50% at t = 180, 260, 320, 420, and 450 s, respectively. All points reach at equilibrium concentration till, tm = 480 s.
Mixing by conduit-type is slower than by fin-type. This result can be considered as follows. The fin-type mixer generates a closed loop locally near the mixer. So, the exchange rate of the fluid between the upper region and the lower region is faster near the fin-type mixer due to the two-sided suction and closed loop. On the other hand, conduit-type mixer produces a wider circulatory pattern along the tank wall and one-sided suction flow occurred only from the bottom of the tank.
The flow characteristics of a mixing tank with conduit-type and fin-type mixers were demonstrated by flow visualization and concentration experiment. The conduit-type mixer generates a radial jet stream from the nozzle due to the centrifugal force. This jet flow produces a spiral rotating jet in a horizontal plane. The mixing time of the conduit-type mixer is longer than that of the fin-type mixer because it requires a longer initial time to produce the wider circulatory flow pattern, which is confirmed by concentration measurement and flow visualization. However, the conduit-type mixer provides better uniform mixing than fin-type mixer.
We would like to show our appreciation to Mr. Yushi Koike, Graduate School of Science and Engineering, Saitama University for his great support on our project.
c Concentration rate [%]
d Mixer diameter [m]
D Vessel diameter [m]
e Measurement error [-]
h Mixer position [m]
H Vessel height [m]
I Image intensity [-]
n Rotation rate [rpm]
t Time [s]
tm Mixing Time [s]
Re Reynolds number
µ Viscosity [Pa∙s]
ρ Density [kg∙m−1∙s−1]
ω Angular velocity [s−1]
*A part of this work was presented at 15th International Conference Fluid Dynamics (ICFD15) in Sendai, 7-9 Nov. 2018.