14 th European Conference on Mixing Warszawa, 10-13 September 2012 2D FLUID DEFORMATION INDUCED BY A ROTATIONAL RECIPROCATING PLATE IMPELLER IN A CYLINDRICAL VESSEL Yoshiyuki Komoda a, Saki Senda a, Hiroshi Takeda b, Yushi Hirata c, Hiroshi Suzuki a a Kobe University, 1-1 Rokkodai, Nada, Kobe, Hyogo 657-8501, Japan b Rflow Co., Ltd., 10-45, Takasago, Soka, Saitama 340-0015, Japan c Osaka University, Yamadaoka, Suita, Osaka 565-0871, Japan komoda@kobe-u.ac.jp Abstract. Inspired by excellent mixing at the moment when the movement of a plate is reversed, we have investigated a novel mixing process where a relatively large plate impeller rotates back and forth slowly in a cylindrical vessel (rotational reciprocating mixing). The visualizations of the streak line in the r-θ plane by experiment and simulation show that fluid deformation is well expressed by connecting the tracer particles being tracked according to the periodically stable velocity field obtained by CFD simulation. Since two pairs of vortex flows having counter rotational motion are generated in each semicircle region separated by the impeller plate, it was found that the fluids in the center and outer part of the vessel are effectively exchanged due to suction and discharge flows. The mixing performance of rotational reciprocation is evaluated by the length of tracer fluid depicted by connecting tracer particles initially located on circle. The tracer line increases its length exponentially with mixing time after the first counterturn of the plate impeller regardless of initial positions. Keywords: Reciprocating motion, CFD, visualization, chaotic mixing, stretching, folding 1. MIXING PROCESS WITH RECIPROCATING MOTION Based on various experiences, we know mixing proceeds rapidly at the moment when an impeller starts to rotate. In other words, unsteady motion of the impeller is one of alternatives to achieve rapid mixing. Nevertheless, most mixing equipment has been operated under a constant rotational speed due to a mechanical. Although some researchers proposed unsteady mixing operations, they are operated under constant rotational speeds with periodic change in rotational direction and speed [1, 2]. On the contrary, we have investigated the mixing process, where a disk impeller goes up and down slowly in a cylindrical vessel with a sinusoidal change in disk speed (vertical reciprocating mixing) [3-5]. Since the disk always accelerates or decelerates during the mixing, the doughnuts-shaped vortex flow developed and then destroyed behind the disk. It was clarified that an effective chaotic mixing was achieved by the phase lag between the vortex flows and the disk impeller. It is easily expected that a rotational reciprocating motion of a plate impeller in a cylindrical vessel will have a unique fluid flow and probably enables to establish a rapid mixing process. As expected from the vertical reciprocating mixing, the 3D mixing process should be too complicated to extract the important factor dominating mixing or fluid deformation. In this study, we have investigated the 2D fluid deformation in the r-θ plane induced by a rotationally reciprocating plate by CFD simulation and experimental observation. 211
2. CALCULATION AND EXPERIMENT 2.1 Calculation of velocity field In the circular cross-section of a cylindrical mixing vessel having an infinitively long height, the 2-D flow induced by a plate impeller reciprocating rotationally was simulated using CFD software (R-FLOW). The schematic figure of the simulated domain is shown in Figure 1. In the reciprocating mixing, the impeller has a role as a baffle while decelerating its speed and a large impeller is considered to be more effective to make the flow fully baffled. The diameters of the vessel and plate impeller were set 0.2 m and 0.15 m, respectively. The impeller is attached to the shaft having the diameter of 0.02 m, which is rotated in the θ direction. The rotational speed was changed as a sinusoidal function. The orientation of the impeller ψ from the starting position is expressed by Eq. (1). ψ = A 1 cos 2πt T (1) [ ( )] where, A and T represent the amplitude and the period of impeller reciprocation. The angle swept by the impeller reciprocation is given by 2A and the average rotational speed N ave is then given by 4A/T. The calculated area was composed of vessel and impeller blocks as shown in Figure 1. While the vessel block was fixed to be stationary, the impeller block located inside the vessel block was rotated with the impeller and shaft. The density of the test fluid, ρ, was 1000 kg/m 3 and the viscosity, μ, was 0.1 Pa s. Since the Reynolds number (Re = ρn ave d 2 /μ) was 113, the laminar flow calculation was carried out. After the calculation for several periods, the periodically stable velocity profiles were obtained. 2.2 Experimental visualization of streak line The streak line of the tracer fluid in the r-θ plane was visualized from the bottom of the vessel by a laser sheet technique. The experimental setup is shown in Figure 2. The diameters of the vessel, plate impeller and shaft were 8cm, 6cm and 8mm, and these dimensions are analogous to those adopted in the CFD simulation. Since the height of the plate impeller is 10cm and the liquid surface was 6cm from the bottom, the fluid flow in the r-θ plane near the liquid surface was roughly two dimensional. The angle of the plate impeller controlled by a stepping motor was same as Eq. (1). An aqueous solution of starch having the constant viscosity of 0.1 Pa s was used as a test fluid and the same solution containing Rhodamine B was as a tracer fluid. After a several periods from the start of reciprocation, the injection of the tracer fluid was started at a constant rate (0.6ml/min) using a syringe pump at various impeller positions. The injection point locates 3.2cm from the center of the vessel and 1cm below the liquid surface. The r-θ plane at the same height was illuminated using a laser sheet in order to visualize a streak line. The development of the streak line was recorded by a digital camera via a mirror located under the bottom of the vessel. 0.2m Rotating impeller block stepping motor syringe pump PC 0.02m laser sheet lihgt source 0.15m Stationary vessel block Figure 1. Dimensions of CFD simulation domain. mirror digital camera Figure 2. Experimental setup for visualization. 212
3. FLUID FLOW GENERATED BY A ROTATIONAL RECIPROCATING PLATE Figure 3 shows the variation of velocity profiles in the acceleration period of the plate impeller. When the impeller starts to rotate, fluid flows into the gap between the impeller tip and vessel wall. As the impeller rotates, fluid behind the impeller tip goes back to the gap region. Consequently, a pair of vortex flows is generated behind the impeller tip. These vortex flows then separate from the impeller tip and are then stretched along the vessel wall. Finally, the vortex flows generated in the last stroke of reciprocation remains at the beginning of the next stroke. Since two pairs of vortex flows thus generated rotate in the counter directions, two pair of suction and discharge flows into and from the center part of the vessel was induced in the semi-circle region separated by the impeller plate. Therefore, the fluids in the center and outer part of the mixing vessel should be exchanged effectively by this mixing method even under a laminar mixing condition. vortex flow generated in the last stroke t/t = 0 t/t = 0.1 vortex flow generated in the present stroke t/t = 0.3 Figure 3. Velocity profiles in r-θ plane generated by a rotationally reciprocating impeller. 4. COMPARISON OF STREAK LINES The coexistence of suction and discharge flows suggests that the fluid near the vessel wall is drawn into the center part of the vessel and again pushed outward to the vessel wall simply by rotationally reciprocating the plate impeller. In order to confirm the motion of fluid in the mixing vessel, the development of the streak line was simulated using periodically stable velocity field as well as experimental observation. A fluid particle is first put at the injection point and tracked according to the velocity field at each time step by FORTRAN program. If the distance between the fluid particle and the injection point exceeds a set value, a new fluid particle was added at the injection point and then tracked in the similar manner. Consequently, the streak line was depicted by connecting these fluid particles. The velocity field used here was calculated under the same condition with the experiment. t/t = 0 0.2 0.4 0.6 0.8 1.0 Figure 4. Comparisons of streak lines between simulation (upper) and experiment (lower). 213
Figure 4 show the development of streak lines obtained by simulation and experiment. The injection of the tracer fluid began at the moment when a quarter period before the impeller counterturn at t/t = -0.25 and ψ = -π/4. In the first stroke of reciprocation (t/t = 0~0.5), it was observed in the experiment that the tracer fluid was drawn along the track of the impeller tip, and the streak line was folded and stretched along the impeller plate. After the movement of the impeller plate was reversed, a part of the streak line near the injection point was again stretched along the track of the other impeller tip. As a result, a similar stretching and folded behavior was observed along the same surface of the impeller plate with the last half stroke. At the same time, the other edge of the streak line remains in the semi-circle region where the streak line was first folded and then stretched along the vessel wall. In this way, the tracer fluid was stretched effectively along either the impeller plate or the vessel wall. The development of the streak line observed in the experiment was perfectly expressed by simulation. In the following section, the characteristics of mixing mechanism were described based on the deformation process by tracking the fluid particles using calculated velocity field. 5. DEFORMATION OF TRACER LINES In the rotational reciprocation the flow pattern accompanied with periodic emergence and destruction of vortex flows is significantly complicated. In this study, the mixing process is visualized by tracking the deformation of circular tracer line. A tracer line is drawn by connecting discrete fluid particles that are tracked in the similar manner with the streak line calculation. In the calculation, if the distance between the adjacent particles becomes larger than the initial value, an additional particle is put at the midpoint between them. On the other hand, if the distance becomes smaller than a half of the initial value, the corresponding particle is removed. In this way, the particles line up at almost regular intervals and satisfactorily express a deformed line. t/t =0 0.2 0.4 0.6 0.8 1.0 Figure 5. Deformation process of circular fluid interfaces (radius are 0.03, 0.05, 0.09 m from top). Figure 5 shows the deformation processes of circular tracer lines having the initial radius of 0.03, 0.05 and 0.09 m, respectively. The tracer line initially located near the shaft (0.03m) was first stretched along the impeller plate and then pushed outward to the impeller tip. A part of the tracer line reached at the impeller tip was successively drawn into the vortex flows behind the impeller tip. Finally, the vortically deformed tracer line fills the semicircle region divided by the impeller plate. As inferred from the deformation process of inner tracer lines, the circular tracer line initially located near the impeller tip (0.05m) was stretched along the 214
track of the impeller tip just after the beginning of impeller reciprocation. Since the tracer line was significantly stretched along the vessel wall at the first impeller counterturn, the tracer line was extensively deformed vortically compared with inner tracer lines. On the contrary, the fluid line near the vessel wall (0.09m) showed a quite different deformation process until it is drawn into vortex flows. The fluid line was first drawn into the center part of the vessel due to suction flows and successively pushed outward along the impeller plate due to discharge flows. Since the tracer line hardly approached the impeller tip, the line did not show significant vortical deformation. Although any tracer line finally showed an effective stretching behavior due to the vortex flows behind the impeller tip, the suction and discharge flows have the most important role to transport fluid to the impeller tip. l(t)/l(t = 0) [-] 1000 100 10 radius of tracer line 0.03m 0.05m 0.07m 0.09m 1 0 1 2 t/t [-] Figure 6. Variation of elongation ratio of circular tracer lines. Assumed that a whole fluid in the vessel was divided by these tracer lines, the tracer line corresponds to the interface between two fluids having the exactly same physical properties. Therefore, the thickness of the fluid decreases as the length of the fluid interface increases. If the fluid becomes sufficiently thin to the order of molecules, these two fluids separated by the tracer line are considered to be completely mixed. Therefore, the line length l(t) is calculated as a summation of the distance between adjacent particles, and its elongation ratio l(t)/l(t=0) is adopted as a measure to evaluate a mixing rate. Figure 6 shows the variation of elongation ratios of circular tracer lines having various initial radii. The tracer line located almost along the track of the impeller tip increase its length exponentially with time just after the beginning of mixing. On the contrary, the tracer line initially located near the shaft increased promptly and that located near the mixing vessel increased slowly in the early stage of mixing. Nevertheless, all of these tracer lines were elongated more than 10 times in every cycle of reciprocation after t/t = 0.5. As can be seen in Figure 5, the deformation of each tracer line becomes to be affected by vortex flows at that time. It also suggests that the fluid element will be thinned to less than 1/10 6 within 5 cycles of reciprocation or 10s in mixing time. 6. NOMENCLATURE A : amplitude of reciprocation [rad] D : inner diameter of the mixing vessel [m] d : diameter of the plate impeller [m] l : length of tracer line [m] T : period of reciprocation [s] ψ : angle of the plate impeller from the starting point [rad] 215
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