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Surface vortex formation and free surface deformation in an unbaffled vessel stirred by on-axis and eccentric impellers
Chemical Engineering Journal 367 (2019) 25–36
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Surface vortex formation and free surface deformation in an unbaffled vessel stirred by on-axis and eccentric impellers
T
⁎
Takuya Yamamoto , Yu Fang, Sergey V. Komarov Graduate School of Environmental Studies, Tohoku University, Miyagi 980-8579, Japan
H I GH L IG H T S
types of transient free surface deformation are observed. • Three surface vortex is developed due to a low pressure-zone behind the impeller blade. • Local rotation of free-surface elevation is generated due to the macro-instability. • Entire free-surface elevation and deflection are generated due to trailing vortices. • Local • The stirring method determines the contribution of each phenomena.
A R T I C LE I N FO
A B S T R A C T
Keywords: Stirring Volume of Fluid (VOF) method OpenFOAM Surface vortex Eccentric stirring Macro-instabilities
The present study investigated the mechanism of surface vortex formation and free surface deformation in an unbaffled stirred vessel with on-axis and eccentric impellers having four paddle blades. To shed light on the mechanism of surface vortex formation and free surface deformation, we conducted experiments and numerical simulation in an unbaffled cylindrical stirred vessel. The surface vortex was found to be small in the eccentrically stirred vessel due to a vertical flow caused by the interaction between the discharged flow and sidewall. In the case of on-axis impeller, low-pressure zones at the rear faces of blades pulled the free surface, that causes the air entrainment. These phenomena could be seen in both the experimental observation and numerical simulation. The three types of transient free surface deformation were observed: local vortex formation due to the lowpressure zone mentioned above, entire rotation of free-surface elevation due to the macro-instabilities, and local elevation and deflection due to trailing vortices.
1. Introduction Mechanical stirring is a crucial operation in many industrial applications, for example in chemical and metallurgical processes [1–3], food [4] and biological engineering [5,6]. In chemical processes, liquid chemicals should be mixed quickly and uniformly to enhance reaction rates. In metallurgical processes, molten metals should be agitated to reach a uniform concentration and to remove impurities from the melt. As a rule, the agitation operation is combined with bubble injection and/or flux treatment. When an impeller is rotated rapidly, the fluid flow becomes turbulent and the free surface shape is deformed. The behavior of the free surface plays a significant role in many processes, particularly those where there is need to enhance mass transfer from the free surface. For example, oxygen has to be transported from the free surface into cell cultures. In metallurgical processes, the movement of free surface can cause of entrainment of oxide films into the melt. ⁎
Therefore, both the stirring efficiency and free surface movement must be properly controlled in the actual processes. In the present study, the free surface vortex occurring during a mechanical agitation of a liquid bath in a tank has been investigated. Hereinafter, this vortex will be termed “surface vortex” to distinguish it from the turbulent vortex. The depth of surface vortex is evaluated by semi-analytical and empirical correlation equations. Below is a brief overview of the recent studies in this area. Nagata [7] was the first who presented a correlation equation for the surface vortex depth in an unbaffled stirred tank. Rieger et al. [8] also introduced a correlation equation between the surface vortex depth and non-dimensional numbers based on an analytical equation derived from the balance between the surface tension force and dynamic pressure. Kamei et al. [9] presented an empirical correlation equation between the surface vortex depth and geometric parameters in a baffled stirred tank. Bhattacharya et al. [10], and Mali and Patwardhan [11] created a correlation
Corresponding author. E-mail address: [email protected] (T. Yamamoto).
https://doi.org/10.1016/j.cej.2019.02.130 Received 10 July 2018; Received in revised form 19 December 2018; Accepted 18 February 2019 Available online 19 February 2019 1385-8947/ © 2019 Elsevier B.V. All rights reserved.
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equation of surface vortex focusing on the turbulent flow velocity near the free surface. Rao et al. [12] and Sato et al. [13] also presented different types of empirical correlation equations. Recently, Deshpande et al. [14] derived a new empirical relationship based on their simulated and experimental results. They proposed a corrected Froude number to generalize the surface vortex depth. Along with studies on correlation equations, the surface vortex depth has been also simulated numerically. Ciofalo et al. [15] were the first who simulated a flow with the free surface deformation in unbaffled tanks stirred by a Rushton turbine and eight-blade paddle impellers. The tracking of free surface shape was carried out using the body fitted coordinate (BFC) system. The simulated results are in good agreement with the results of Nagata [7]. Serra et al. [16] conducted a numerical simulation for a stirring tank with four baffles using the Volume of Fluid (VOF) method. To express the turbulent flow in a stirred tank, the Reynolds averaged Navier-Stokes (RANS) equation was solved. The Reynolds stress in the RANS equation was modeled by a standard k-ε model. After this research, almost all authors adopted the VOF method for free-surface simulation because the VOF method can capture well the complex deformation of the free surface. It has been found that the calculation results are strongly affected by the turbulence model used for the simulation. Therefore, the validation and verification of flow structures, predicted by a given turbulence model, has been one of the main subjects of recent investigations. Glover et al. [17] carried out a numerical simulation to capture the free surface dynamics using the VOF method. The RANS equation was used for the turbulent model and k-ω SST model was adopted to simulate the Reynolds stress. Haque et al. [18] evaluated the numerical accuracy regarding the modeling of Reynolds stress in an unbaffled stirred tank. They compared the k- ω SST and the Reynolds Stress Transport (RST) models. Mahmud et al. [19] also compared the k- ω SST and RST models used to simulate fluid flows in an unbaffled tank stirred by a magnetic stirrer. In these studies, the RST models could predict the flow field accurately. Haque et al. [20] investigated the calculation accuracy by changing Reynolds stress models. The authors compared the k- ε, the k- ω SST and the RST models developed by Launder et al. [21] (RSTLLR) and by Speziale et al. [22] (RST-SSG). The RST models could predict the turbulence flow field better than the other ones. However, the above models were incapable of predicting the turbulent kinetic energy well because they used the ensemble-averaged equation and modeled Reynolds stress. Yang and Zhou [23] adopted the Detached Eddy Simulation (DES) for the free surface turbulent flow in an unbaffled stirred tank. The DES model could predict the turbulent flow better than the turbulent models using the RANS equation. Lamarque et al. [24] adopted the Large Eddy Simulation (LES) instead of RANS equation to simulate the flow in an unbaffled vessel stirred by a magnetic stirrer. Deshpande et al. [14] adopted the LES scheme for an unbaffled stirred tank. They carefully validated the shape of vortex formation. The above studies have found the following. (1) The LES scheme can predict well the turbulent bulk and free surface flows in stirred vessels, although the calculation cost is enormous. (2) The RST model is a better choice to predict the turbulent flows when one solves the RANS equation. (3) The DES scheme is another choice to predict the turbulent flows with a good degree of accuracy. The development of experimental methods to measure the exact free surface shape in a stirred tank is another research topic. Generally, a comparison of free surface shape between experimental observations and numerical simulation is quite difficult. Torre et al. [25] compared the time-averaged grey-scaled photographs with simulated results using the Eulerian-Eulerian multiphase model in a baffled stirred vessel. The calculated interface shape was predicted by a contour surface corresponding to the calculated volume fraction of 0.9. Torre et al. [26] also investigated the transient free surface behavior soon after the agitation was stopped in the same stirred vessel as that in their previous study [25]. Busciglio et al. [27] investigated the free surface shape by taking photographs, which were back-illuminated by two lamps. The
photographs were overlapped to obtain the time-averaged free surface shape. Busciglio et al. [28] also investigated the oscillation characteristics of free surface in an unbaffled stirred tank. The oscillation frequency was found to be different from the impeller rotational frequency, and was dependent on the vessel geometry and impeller rotation speed. The free surface movement has been also carefully investigated. Jahoda et al. [29] studied the surface oscillation in a baffled stirred tank through a numerical simulation. The simulation used a LES scheme for the turbulent modeling and a VOF model for the interface capturing. They concluded that the surface oscillation is caused by a macro-vortex located near the free surface. Kulkarni et al. [30] investigated gas entrainment through the VOF-LES scheme. According to their results, small cavity formed at the free surface is a source of gas entrainment. Fluid strain rate in the gas phase becomes stronger when the surface aeration occurs. Yang et al. [31] numerically investigated the free surface dynamics in an eccentrically stirred tank. As Yang et al. [31] discussed, the turbulent flow and free surface movement in an eccentrically stirred vessel are different from those in an on-axis stirred vessel. Nishikawa et al. [32] were the first who reported this finding. An eccentrically located impeller can reduce the mixing time, and the location of the surface vortex is changed. Hall et al. [33,34] compared an eccentric stirring in an unbaffled vessel with a centered stirring in a baffled vessel concerning the mixing time and the distribution of turbulent kinetic energy, which is one of the indicators of assessing the mixing efficiency. They indicated that the turbulent kinetic energy spreads widely in the eccentric stirring, that improves the mixing efficiency. Nomura and Iguchi [35] made experiments using an eccentric impeller to observe the shape of the surface vortex. They found that the developed surface vortex becomes inclined, and many low-density particles can be entrained from the surface vortex. Ohmi et al. [36] classified the shape of the surface vortex in an unbaffled cylindrical vessel stirred by an eccentric impeller. Galletti et al. [37] investigated the frequency of flow velocity oscillations in an unbaffled vessel agitated by an eccentrically located impeller. It has been found that the macro-instabilities is the main cause of these oscillations, and the Strouhal number can describe the variation of the oscillation frequency. Galletti et al. [38] also investigated the angle of the surface vortex inclination and the mixing time using a de-colorization experiment in an unbaffled vessel stirred by an eccentric impeller. Liu et al. [39] investigated bubble distribution in an eccentrically stirred unbaffled vessel with gas injection. The surface vortex shape and flow characteristics were found to be quite different from those in an on-axis stirred vessel. As discussed above, the dynamic behavior of the free surface during both on-axis and eccentric stirrings have been not clarified properly especially for the transient movement of the free surface. To shed light on the three-dimensional behavior of the free surface, we investigated the mechanisms of free surface deformation and vortex formation by conducting experiments and numerical simulation in an unbaffled vessel stirred by an on-axis or eccentric impeller. The main emphasis was placed on the transient structure of bubbles entrained from the turbulent free surface, and relationships between the dynamic behavior of the free surface and the flow structures. 2. Experimental procedures The configuration of the experimental system is shown in Fig. 1. An unbaffled cylindrical vessel was filled with water agitated by a paddle impeller with four thick blades. The blade tip-to-tip diameter d was 60 mm, the initial water level h was 320 mm, the inner diameter, D and height, H of cylindrical vessel was 192 and 400 mm (d = 0.3125 D; h = 1.667 D; H = 2.08 D), respectively. The distance between the initial water level h and the upper surface of blades was 40 mm. The clearance between the impeller midplane and the bottom wall, C was 274 mm (C = 1.427 D). The shaft diameter, ds, the impeller height, hI, 26
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lamp. The free surface shape was precisely measured by the following way. The halogen lamp was switched on to irradiate strong backlight, which was scattered and/or attenuated at the free surface. Pictures taken by the camera were blacked out at the location of the free surface. To obtain the oscillation amplitude and time-averaged profile of the free surface, the images were processed as shown in Fig. 3. First, the taken pictures were time-averaged for 2 s (1000 photos). Second, the difference in images between the time-averaged pictures and instant pictures was evaluated for each picture. Third, thus obtained picture was time-averaged for 2 s. Finally, the color of the time-averaged picture was inverted to better observe the picture details, as shown in Fig. 3. However, the free surface shape obtained by this procedure is assumed to over-estimate the free surface deformation because the backlight is scattered and/or attenuated at the projected area. Additionally, the transient free surface shape was obtained in a different technique. Strong light was irradiated from the upper side of stirring vessel to highlight the transient free surface shape. 3. Numerical methods and procedures 3.1. Governing equations for a liquid-gas flow The governing equations are the following balance equations of momentum and mass:
∂ (ρu ) + ∇ ·(ρuu ) = −∇p + ∇ ·μ (∇u + ∇uT ) + ρg + Fσ ∂t
(1)
∂ρ + ∇ ·(ρu ) = 0 ∂t
(2)
where ρ is the fluid density, u is the flow velocity, t is time, p is pressure, μis the viscosity of the fluid, and Fσ is the surface tension term. The present study adopted the Volume of Fluid (VOF) method (e.g. [40–42]) to capture the liquid-gas interface. The VOF method has many modifications to calculate the dynamics of free surface accurately and/ or easily. Among them, an algebraic VOF method has the advantage due to convenience to simulate multiphase flows in complicated geometry. Therefore, the algebraic VOF method was used in the present study. The algebraic VOF method developed by Weller [40] is expressed as:
Fig. 1. A schematic representation of unbaffled stirring vessel used in the present study.
and the impeller width, wI was 24 mm, 12 mm, and 6 mm, respectively. The impeller was rotated at constant rates of 300 rpm and 400 rpm. The distance between the shaft axis line and the vessel axis line was 0 and 60 mm. Hereinafter, this distance will be termed “eccentric length”. The driven flow was characterized by two non-dimensional parameters, Reynolds number (Re = nd2/ν) and Froude number (Fr = n2d/g), where n is the impeller rotation speed [rps], ν is the kinematic viscosity [m2s−1], g is the gravitational acceleration [ms−2]. The Reynolds number and Froude number are equal to 18,000 and 0.153 for 300 rpm rotation speed, and 24,000 and 0.272 for 400 rpm rotation speed, respectively. The free surface shape was recorded by a high-speed camera as shown in Fig. 2. The camera was fixed at the opposite side of a halogen
∂α + ∇ ·(αu ) + ∇ ·((1 − α ) αur ) = 0 ∂t
(3)
where α is the volume fraction of liquid and ur is the relative velocity between the liquid and gas phases. The definition of volume fraction α is:
α=
Liquid 1 0 Gas ⎨0 < α < 1 Interface ⎩ ⎧
(4)
In the present study, the contour of α = 0.5 depicted the free surface. The physical properties are updated according to the volume fraction α as:
ρ = ρl α + ρg (1 − α )
(5)
μ = μl α + μg (1 − α )
(6)
The relative velocity ur is treated as the compressive velocity. The third term on the left-hand side in Eq. (3) is discretized as:
∫Ω ∇·((1 − α ) αur ) dΩ
= ∑ ((1 − α ) αur )·Sf = ∑ ((1 − α ) αϕr )f
(7)
where Sf is the grid cell surface vector, ϕr is the volumetric compressive flux, and the subscript f is the cell face. The volumetric compressive flux ϕr is expressed as:
Fig. 2. Schematic representation of shooting method to obtain the free surface shape. 27
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Fig. 3. Procedure of image analysis: (a) an instant photograph, (b) a time-averaged picture, (c) a subtracted picture, (d) a time-averaged subtracted picture, and (e) a color-inverted picture.
where σ is the surface tension and k is the curvature of liquid-gas interface. Readers interested in more details on curvature calculation are referred to Rusche [41] and Yamamoto et al. [42].
the computational cost. The time step for the advection equation of liquid fraction was split and solved in the split time step. The number of sub-cycle iteration is 5. The temporal derivatives were discretized by a second order backward scheme except for Eq. (3). The multi-dimensional limiter for the explicit solution (MULES) scheme (e.g. [41]) was adopted for the temporal discretization of Eq. (3). The spatial discretizations are the second order linear interpolation except for Eq. (3). One of the TVD schemes, namely the second order linear-upwind scheme was used to avoid the numerical oscillations which can occur because the volume fraction in Eq. (3) has a steep gradient at the liquid-gas interface. The velocity and pressure fields were implicitly coupled by the PIMPLE algorithm, which is an improved version of PISO algorithm [44]. The OpenFOAM version utilized is 2.3.x.
3.2. Numerical procedures
3.3. Numerical grid resolution
The turbulence model used is the Large Eddy Simulation (LES) scheme. The solved equations in LES model are changed into
The numerical grid resolution was determined based on the Taylors micro-scale in the same way as Kulkarni and Patwardhan did [30]. They suggested that the Taylor micro-scale in a vessel agitated by an impeller with paddle blades is approximately 1–2 mm. Therefore, a numerical grid, the size of which near the impeller and the free surface is smaller than 1 mm, was used in the present study. Also, the Taylor micro-scale was estimated. In the present study, the turbulent energy dissipation rate ∊ was calculated as:
|ϕf | ⎤ ⎞ ⎛ |ϕf | n (ϕr )f = min ⎜Cα , max ⎡ ⎢ ⎥⎟ f |Sf | ⎣ |Sf | ⎦ ⎠ ⎝
(8)
where C α is the compression adjusting parameter. This parameter can change the degree of surface compression. The present study used unity for the compression adjusting parameter. The related numerical models are described in details in the relevant literature [40–42]. The surface tension force Fσ in Eq. (1) is set using the continuum surface force (CSF) model:
Fσ = σk∇α
(9)
∂ (ρτij ) ∂p¯ ∂ (ρu¯i ) ∂ ⎛ ∂u¯i ⎞ ∂ (ρu¯i u¯j ) = − + − + ⎜μ ⎟ + Fi ∂x j ∂x j ⎝ ∂x j ⎠ ∂x i ∂t ∂x j
(10)
∂ (ρu¯i ) ∂ρ =0 + ∂x i ∂t
(11)
where τ is the sub-grid scale stress tensor and the overlines indicate the filtered variables. The Sub-Grid Scale (SGS) turbulent kinetic energy is calculated on the basis of the transport equation of turbulent kinetic energy. The chosen cut-off filter is determined by the cubic root of grid cell volume (cubeRootVol). One-k-equation eddy viscosity model (oneEqEddy) is adopted for the sub-grid scale viscosity. The sliding grid technique directly expresses the rotation of the impeller. An open source software, OpenFOAM [43], was used for the present simulation. The sliding technique with the algebraic VOF method was simulated by interDymFoam, which is a solver implemented in OpenFOAM. The solver can handle the adjustable time step technique to reduce the calculation time. The time step was controlled using the local Maximum Courant number. In the present research, the Maximum Courant number was set to be 0.2. To capture the interface shape adequately, the solver adopted the sub-cycle algorithm for the advection equation of liquid fraction. When one utilizes this algorithm, the temporal resolution of free surface shape becomes high with the slight increase in
∊ = 2νSGS S¯ij S¯ij
(12)
where νSGS is the sub-grid-scale (SGS) viscosity, and Sij is the rate-ofstrain tensor. This calculation method is the same as that in the study of Murthy and Joshi [45]. The maximum turbulent energy dissipation rates are around 2–3 m2s−3 for 300 rpm rotation and around 5 m2s−3 for 300 rpm rotation. The Taylor micro-scale λ is expressed as:
λ2 =
15νu2 ∊
(13)
Finally, the Taylor micro-scale was estimated to be 2–3 mm for both the 300 rpm and 400 rpm conditions. Therefore, our calculation can resolve the turbulent vortices, the size of which is larger than the Taylor micro-scale. The vortices smaller than the Taylor micro-scale are modeled through the filtering used in a LES scheme. Finally, the total number of numerical grid cells was about 4,000,000. This suggests that the grid resolution used is sufficient to investigate the bulk and free 28
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surface flow dynamics through the LES scheme.
shape of the free surface is expressed as the time-averaged volume fraction α:
3.4. Boundary conditions
α¯ = At the sidewall and impeller surface, no-slip condition for velocity and zero gradient condition for volume fraction were imposed. At the top boundary, the flow velocity was determined by pressure distribution (pressureInletOutletVelocity), and the volume fraction was changed depending on the flow direction (inletOutlet). When the flow direction was toward the inner side of the numerical domain, the liquid fraction was set to zero. On the other hand, the boundary condition for liquid fraction was zero gradient to the boundary. Along the sliding grid interface, the cyclic arbitrary mesh interface (AMI) boundary condition was adopted to link the rotating and stationary boundaries. The residual error of algebraic matrix solver was set to be 1 × 10−6. The computation was carried out using the super-computing system in Kyoto University. The machine used was Cray CS400 2820XT, equipped with Intel Xeon Broadwell processors. It took about 4 weeks to complete simulation for a 15 s stirring by using 144 processors.
∫0
t'
αdt
(14)
where the t′ is the time-averaging interval, which is set to 5 s in the present study. The deviated free surface shape is expressed as the rootmean-square (RMS) value of volume fraction α:
α rms =
(α − α¯)2
(15)
The free surface deformation is increased with impeller rotation speed as seen in Fig. 4. This phenomenon is obvious because the turbulent intensity is enhanced with an increase of the impeller rotation speed. It is seen that the deformation magnitudes of simulated free surface shape are slightly smaller than those observed in the experiments. The possible reason for this difference is originated from the measurement method and calculation condition. As discussed in Section 2, the experimentally observed free surface corresponds to the projected area irradiated by light. The projected area involves the threedimensional free surface shape, which can be a source of the over-estimated deformation of the free surface. The calculation condition also can cause this difference. Time-averaging the free surface shape was started in 10 s after beginning the agitation. As Deshpande et al. (2017) [14] showed, the power number is converged to a specific value after about 20 impeller revolutions, which correspond to a time of 6–7 s in the present study. It was confirmed that the power number approaches a certain value within 10 s, although the surface vortex might not be fully developed yet. The depth of not fully developed vortex becomes smaller and the free surface is inclined at a lower angle compared to that at the fully developed vortex. Thus, insufficient number of impeller revolutions might cause this discrepancy. However, since this discrepancy is small, agreement between the experimental and numerical free surface deformations can be considered quite satisfactory. Finally, the experimental and numerically predicted transient free surface shapes were also compared. Fig. 5 shows the snapshots of surface vortex for the case of on-axis agitation at 400 rpm. Fig. 6 shows the time variation of simulated free surface shape near the on-axis impeller at the same rotation speed. In the both cases, small bubbles are entrained from the free surface following a number of steps: (1) The local surface vortex is elongated toward the impeller blade; (2) the local surface vortex reaches the rear surface of blade; (3) small bubbles are detached from the free surface; (4) the bubbles are moved on to the rear surface of blade; (5) the free surface is forced to move up by the next
4. Results and discussion 4.1. Comparison between experimental and numerical simulation results The main emphasis of present study was placed on the surface vortex formation and free surface deformation. Therefore, validation of the numerical model was carried out by comparing the free surface shapes obtained experimentally and numerically. First, the threshold of bubble entrainment was investigated depending on the impeller rotation speed in both the experiments and numerical simulation. In the experiments, the rotation speed was increased in an increment of 10 rpm. The experiments show that the critical rotation speeds of bubble entrainment for the on-axis and eccentric impellers are 390 rpm and 460 rpm, respectively. In the numerical simulation, air is entrained only in the case of the on-axis impeller at 400 rpm, while there is no air entrainment in the cases of the on-axis impeller at 300 rpm and the eccentric impeller at 300 rpm and 400 rpm. Therefore, the experimental and numerical conditions of bubble entrainment are in qualitatively agreement with each other. Next, the experimental and numerical free surface shapes and their oscillation magnitudes were compared. Deviation of the free surface shape from the time-averaged shape is shown in Fig. 4 for the stirring by on-axis impellers at 300 and 400 rpm. The calculated time-averaged
Fig. 4. The free surface shapes deviated from the time-averaged shapes when the liquid bath was stirred by the on-axis impellers of 300 and 400 rpm: (a) Experimental results, and (b) simulated results. 29
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Fig. 5. Experimentally obtained snapshots of surface vortex in the case of on-axis agitation of 400 rpm rotation.
Fig. 6. Time variation of simulated free surface shape near the on-axis impeller of 400 rpm rotation. 30
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Fig. 7. Side views of time-averaged free surface shape: (a) On-axis agitation of 300 rpm, (b) on-axis agitation of 400 rpm, (c) eccentric agitation of 300 rpm, and (d) eccentric agitation of 400 rpm.
approaching blade. These five steps are repeated. The similar time variation of the free surface movement could be observed in the numerical results as seen in Fig. 6. These findings suggest that the transient free surface shapes obtained by experiments and numerical simulation are in good agreement with each other. Thus, the above consideration confirms the validity and accuracy of the numerical model in terms of both the time-averaged and transient structures of the free surface. Therefore, the flow mechanisms and relationship between the free surface deformation and flow structures will be discussed on the basis of the simulated results in the following sections.
stirring conditions. The free surface deflection occurs at the point C in Fig. 8(c) and (d). A downward flow near the free surface causes this phenomenon. The liquid lifted by the upward flow at the narrow space must flow down somewhere to satisfy the conservation of mass. The downward flow is generated immediately after the fluid passes through the narrow space as shown in Fig. 9(b) and (d). 4.3. Transient flow characteristics 4.3.1. Local surface vortex As shown in Figs. 5 and 6, the local surface vortex is generated at the free surface. The generation mechanism of this vortex is discussed in this subsection. Fig. 10 depicts the free surface shape and pressure distribution near the impeller blades. The pressure is high in front of the impeller blades and low behind them. The low-pressure zone near the rear surfaces of blades forces the free surface to shift into a deeper place. Also, the high-pressure zone in front of the next approaching blade pushes the free surface up. Therefore, these results suggest that the pressure distribution around the rotating impeller is responsible for the local surface deflection.
4.2. Time-averaged flow characteristics First, the time-averaged free surface shapes formed in the on-axis and eccentric agitations were compared. Fig. 7 shows side views of the time-averaged free surface shape. The free surface was numerically defined as a surface where the contour of the time-averaged volume fraction equals to 0.5. The interface deformation becomes large with increasing the impeller rotation speed. And this effect is greater for the on-axis agitation than that for the eccentric agitation. The time-averaged free surface shapes are axisymmetric in the former case and nonaxisymmetric in the latter case. Fig. 8 shows top views of the timeaveraged free surface shape. In the case of the eccentric impeller, the free surface is elevated at the narrow zone between the impeller and the sidewall. To clarify the mechanism of these phenomena, the timeaveraged flow pattern was investigated. Fig. 9 shows the cross-sectional velocity vectors produced by the eccentric impellers. The white line indicates the cross-sectional free surface in Fig. 9. The discharged liquid flows and collides with the sidewall forcing the flow to change its direction into upward one. This upward flow elevates the free surface at the narrow space between the impeller and sidewall. As a result, the elevated part of the free surface (hereinafter referred to as “elevation”) located at the same distance from the sidewall as seen in Fig. 8(c) and (d). Also, this upward flow is assumed to decrease the free surface deformation making the air entrainment from the free surface difficult compared with the on-axis
4.3.2. Entire rotation of the free surface elevations The transient deformation of the free surface has been also investigated. The time variation of free surface shape agitated by the onaxis impeller at 400 rpm is shown in Fig. 11. The free surface elevations are located at two places and slowly rotated with a speed differing from that of impeller rotation. The rotation of impeller is 4/3 turn for 0.2 s, while that of surface elevations is about 2/5 for the same time interval. Besides, the number of free surface elevations is different from that of impeller blades which equals 4. It means that there is a complex relationship between the behavior of these elevations, the impeller geometry and rotation speed. Therefore, the fluid velocity near the impeller was considered more carefully. The cross-sectional velocity vectors are shown in Fig. 12. The large vortex is generated above the discharged flow, which does not develop toward the side wall straightly because of the turbulent oscillations. This oscillating discharged flow produces the large vortex oscillations, which affect the free surface movement. The 31
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Fig. 8. Top views of time-averaged free surface shape: (a) On-axis agitation of 300 rpm, (b) on-axis agitation of 400 rpm, (c) eccentric agitation of 300 rpm, and (d) eccentric agitation of 400 rpm.
Fig. 9. Time-averaged cross-sectional velocity vectors produced by the eccentric impellers: (a) 300 rpm, viewed from point A, (b) 300 rpm, viewed from point B and (c) 400 rpm, viewed from point A, and (d) 400 rpm, viewed from point B (Points A and B are shown in Fig. 8). 32
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Fig. 10. Time variation of free surface shape and pressure distribution near the on-axis impeller at a rotation speed of 400 rpm.
different from the on-axis stirring case. It means that the macro-instabilities do not occur at all or occurs in a small extent when the impeller is installed eccentrically. In this case, the free surface movement is characterized by small lifting and random deflection. The mechanism of these phenomena is explained below. The free surface behavior of eccentric 300 rpm agitation is similar. The rotation speed affects the magnitude of surface deflection and elevation. In the case of low-speed impeller rotation, the amplitude of free surface oscillation becomes smaller.
slow oscillation of this large vortex is generally referred as to macroinstabilities (MIs). Therefore, the macro-instabilities of large vortex is assumed to cause the rotation of two free surface elevations as shown in Fig. 12. In the case of the on-axis impeller rotating at 300 rpm, the macro-instabilities and related surface elevations are also generated. To characterize the macro-instabilities, the frequency of macro-instabilities per rotation speed (f/N) has been investigated in the previous studies. In the present study, the f/N value of free surface was around 0.22. In the study of Galletti et al., [38], the f/N value was about 0.17 with an on-axis Ruston turbine impeller. The f/N values are in good agreement with each other. This result also indicates that the entire rotation of free surface elevation is caused by the macro-instabilities. Fig. 13 shows the time variations of free surface shape when the bath is agitated by the eccentric impeller at 400 rpm. In this case, observations showed that the characteristics of free surface movement are
4.3.3. Trailing vortices Next, the structure of trailing vortices developed from the rear surfaces of blades was investigated. Fig. 14 indicates the trailing vortices in both the cases of on-axis and eccentric stirring. The vortex structure is depicted using contours of the second invariant of rate-of-
Fig. 11. Time variations of free surface shape of the bath agitated by the on-axis impeller of 400 rpm. 33
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Fig. 12. Transient cross-sectional velocity vectors near the blades in the case of on-axis impeller of 400 rpm: Broken line indicates the large vortex developed above the strong discharge flow.
As seen from the figure, the trailing vortices are elongated in the backward direction. The trailing vortices are thin, and they move slightly upward. Generally, counter-rotating vortex pairs are generated from the rear faces of paddle blades, and the size and shape of vortices are similar. However, in the present study, the blue-colored trailing vortices, generated at the upper half of rear blade, are much longer and stronger compared to red-colored vortices, generated at the lower half of read blade. Such a structure of trailing vortices is assumed to be a
strain tensor [46,47], which is expressed as:
ω¯ 2 ⎞ 1 Q = − ⎛S¯ij S¯ij − 2⎝ 2 ⎠ ⎜
⎟
(16)
where Q is the second invariant of rate-of-strain tensor, which is called Q-criterion, ω is the vorticity vector. The color of the vortex in this figure indicates rotating direction depicted by the circumferential component of vorticity. The depicted value of Q is 5000 s−2.
Fig. 13. Time variations of free surface shape in the water bath agitated by the eccentric impeller at 400 rpm. 34
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low-pressure zone generated behind the impeller blades, (2) entire rotation of the free surface due to the macro-instabilities, and (3) local free surface elevation and deflection at a location between the shaft and side wall due to the trailing vortices. The stirring method (on-axis or eccentric) determines the contribution of each of these phenomena to the surface deformation. Moreover, both the entire rotation of freesurface elevation and the local surface vortex formation affect the bubble entrainment. The bubbles are assumed to be entrapped in the following three steps: (1) The entire rotation of free surface elevations occurs due to the macro-instabilities. (2) The distance between the free surface and impeller blade becomes smaller because the free surface deflection is enhanced near the impeller shaft due to the entire rotation of free surface. (3) The bubble entrainment occurs because the free surface is pulled into the low-pressure zone behind the impeller blade generated by the impeller rotation. 5. Conclusion In the present study, the mechanism of surface vortex formation and free surface deformation in an unbaffled cylindrical vessel stirred by an on-axis or eccentric impeller was investigated. The experimental observations and numerical simulation were performed to clarify these phenomena. It is shown that the results of experiments and simulation are in good agreement with each other. The results of the present study can be summarized as follows:
• The deformation of free surface is small in the case of eccentric • • • •
Fig. 14. Structure of trailing vortices developed near the impeller rotating at 300 rpm and 400 rpm viewed from the side and the top: (a) on-axis stirring, and (b) eccentric stirring.
cause of the discharged flow, which is developed slightly upward due to the pressure distribution. The pressure between the impeller top and the free surface becomes lower because of the strong discharge flow and existence of the free surface. This low-pressure zone causes the discharged fluid to flow slightly upward. As a result, the trailing vortices reach the free surface and affect its shape. Therefore, the trailing vortices are assumed to be responsible for the free surface lifting and deflection at an intermediate distance between the side wall and shaft as seen from Fig. 11. In the case of eccentric stirring, the trailing vortices have short and complex geometry. They are rolled in the narrow gap between the impeller and the side wall. The structures of trailing vortices are changed and shortened at the narrow gap because the flow becomes complicated. The discharged flow transports the vortices forcing them to reach the free surface. Therefore, such vortices produce small and complicated elevation and deflection of the free surface. This is quite different from the large pseudo-periodic flow structure (macro-instabilities), which causes the slow rotation of surface elevations and deflections in the case of the on-axis stirring. The time-averaged free surface deformation has been discussed in terms of Froude number (Nagata [7], Rieger et al. [8], Deshpande et al. [14]). Froude number denotes the balance of inertial and gravitational forces which are significant for the free surface deformation. On the other hand, many factors affect the transient free surface deformation as discussed above. The following three phenomena are responsible for the surface deformation. (1) Local surface vortex formation due to a
•
stirring due to the upward flow developed at the narrow space between the impeller and sidewall. Three types of transient free surface deformation are observed: (1) local surface vortex, (2) entire rotation of free-surface elevation, and (3) local elevation and deflection. The low-pressure zone generated behind of impeller blades is responsible to the formation of local surface vortex. The entire rotation of free-surface elevations is caused by the macroinstabilities. In the case of eccentric stirring, the entire rotation of free-surface elevations occurs to a much smaller extent compared to the on-axis stirring case. The trailing vortices cause the local free-surface elevation and deflection.
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Chemical Engineering Journal journal homepage: www.elsevier.com/locate/cej
Surface vortex formation and free surface deformation in an unbaffled vessel stirred by on-axis and eccentric impellers
T
⁎
Takuya Yamamoto , Yu Fang, Sergey V. Komarov Graduate School of Environmental Studies, Tohoku University, Miyagi 980-8579, Japan
H I GH L IG H T S
types of transient free surface deformation are observed. • Three surface vortex is developed due to a low pressure-zone behind the impeller blade. • Local rotation of free-surface elevation is generated due to the macro-instability. • Entire free-surface elevation and deflection are generated due to trailing vortices. • Local • The stirring method determines the contribution of each phenomena.
A R T I C LE I N FO
A B S T R A C T
Keywords: Stirring Volume of Fluid (VOF) method OpenFOAM Surface vortex Eccentric stirring Macro-instabilities
The present study investigated the mechanism of surface vortex formation and free surface deformation in an unbaffled stirred vessel with on-axis and eccentric impellers having four paddle blades. To shed light on the mechanism of surface vortex formation and free surface deformation, we conducted experiments and numerical simulation in an unbaffled cylindrical stirred vessel. The surface vortex was found to be small in the eccentrically stirred vessel due to a vertical flow caused by the interaction between the discharged flow and sidewall. In the case of on-axis impeller, low-pressure zones at the rear faces of blades pulled the free surface, that causes the air entrainment. These phenomena could be seen in both the experimental observation and numerical simulation. The three types of transient free surface deformation were observed: local vortex formation due to the lowpressure zone mentioned above, entire rotation of free-surface elevation due to the macro-instabilities, and local elevation and deflection due to trailing vortices.
1. Introduction Mechanical stirring is a crucial operation in many industrial applications, for example in chemical and metallurgical processes [1–3], food [4] and biological engineering [5,6]. In chemical processes, liquid chemicals should be mixed quickly and uniformly to enhance reaction rates. In metallurgical processes, molten metals should be agitated to reach a uniform concentration and to remove impurities from the melt. As a rule, the agitation operation is combined with bubble injection and/or flux treatment. When an impeller is rotated rapidly, the fluid flow becomes turbulent and the free surface shape is deformed. The behavior of the free surface plays a significant role in many processes, particularly those where there is need to enhance mass transfer from the free surface. For example, oxygen has to be transported from the free surface into cell cultures. In metallurgical processes, the movement of free surface can cause of entrainment of oxide films into the melt. ⁎
Therefore, both the stirring efficiency and free surface movement must be properly controlled in the actual processes. In the present study, the free surface vortex occurring during a mechanical agitation of a liquid bath in a tank has been investigated. Hereinafter, this vortex will be termed “surface vortex” to distinguish it from the turbulent vortex. The depth of surface vortex is evaluated by semi-analytical and empirical correlation equations. Below is a brief overview of the recent studies in this area. Nagata [7] was the first who presented a correlation equation for the surface vortex depth in an unbaffled stirred tank. Rieger et al. [8] also introduced a correlation equation between the surface vortex depth and non-dimensional numbers based on an analytical equation derived from the balance between the surface tension force and dynamic pressure. Kamei et al. [9] presented an empirical correlation equation between the surface vortex depth and geometric parameters in a baffled stirred tank. Bhattacharya et al. [10], and Mali and Patwardhan [11] created a correlation
Corresponding author. E-mail address: [email protected] (T. Yamamoto).
https://doi.org/10.1016/j.cej.2019.02.130 Received 10 July 2018; Received in revised form 19 December 2018; Accepted 18 February 2019 Available online 19 February 2019 1385-8947/ © 2019 Elsevier B.V. All rights reserved.
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equation of surface vortex focusing on the turbulent flow velocity near the free surface. Rao et al. [12] and Sato et al. [13] also presented different types of empirical correlation equations. Recently, Deshpande et al. [14] derived a new empirical relationship based on their simulated and experimental results. They proposed a corrected Froude number to generalize the surface vortex depth. Along with studies on correlation equations, the surface vortex depth has been also simulated numerically. Ciofalo et al. [15] were the first who simulated a flow with the free surface deformation in unbaffled tanks stirred by a Rushton turbine and eight-blade paddle impellers. The tracking of free surface shape was carried out using the body fitted coordinate (BFC) system. The simulated results are in good agreement with the results of Nagata [7]. Serra et al. [16] conducted a numerical simulation for a stirring tank with four baffles using the Volume of Fluid (VOF) method. To express the turbulent flow in a stirred tank, the Reynolds averaged Navier-Stokes (RANS) equation was solved. The Reynolds stress in the RANS equation was modeled by a standard k-ε model. After this research, almost all authors adopted the VOF method for free-surface simulation because the VOF method can capture well the complex deformation of the free surface. It has been found that the calculation results are strongly affected by the turbulence model used for the simulation. Therefore, the validation and verification of flow structures, predicted by a given turbulence model, has been one of the main subjects of recent investigations. Glover et al. [17] carried out a numerical simulation to capture the free surface dynamics using the VOF method. The RANS equation was used for the turbulent model and k-ω SST model was adopted to simulate the Reynolds stress. Haque et al. [18] evaluated the numerical accuracy regarding the modeling of Reynolds stress in an unbaffled stirred tank. They compared the k- ω SST and the Reynolds Stress Transport (RST) models. Mahmud et al. [19] also compared the k- ω SST and RST models used to simulate fluid flows in an unbaffled tank stirred by a magnetic stirrer. In these studies, the RST models could predict the flow field accurately. Haque et al. [20] investigated the calculation accuracy by changing Reynolds stress models. The authors compared the k- ε, the k- ω SST and the RST models developed by Launder et al. [21] (RSTLLR) and by Speziale et al. [22] (RST-SSG). The RST models could predict the turbulence flow field better than the other ones. However, the above models were incapable of predicting the turbulent kinetic energy well because they used the ensemble-averaged equation and modeled Reynolds stress. Yang and Zhou [23] adopted the Detached Eddy Simulation (DES) for the free surface turbulent flow in an unbaffled stirred tank. The DES model could predict the turbulent flow better than the turbulent models using the RANS equation. Lamarque et al. [24] adopted the Large Eddy Simulation (LES) instead of RANS equation to simulate the flow in an unbaffled vessel stirred by a magnetic stirrer. Deshpande et al. [14] adopted the LES scheme for an unbaffled stirred tank. They carefully validated the shape of vortex formation. The above studies have found the following. (1) The LES scheme can predict well the turbulent bulk and free surface flows in stirred vessels, although the calculation cost is enormous. (2) The RST model is a better choice to predict the turbulent flows when one solves the RANS equation. (3) The DES scheme is another choice to predict the turbulent flows with a good degree of accuracy. The development of experimental methods to measure the exact free surface shape in a stirred tank is another research topic. Generally, a comparison of free surface shape between experimental observations and numerical simulation is quite difficult. Torre et al. [25] compared the time-averaged grey-scaled photographs with simulated results using the Eulerian-Eulerian multiphase model in a baffled stirred vessel. The calculated interface shape was predicted by a contour surface corresponding to the calculated volume fraction of 0.9. Torre et al. [26] also investigated the transient free surface behavior soon after the agitation was stopped in the same stirred vessel as that in their previous study [25]. Busciglio et al. [27] investigated the free surface shape by taking photographs, which were back-illuminated by two lamps. The
photographs were overlapped to obtain the time-averaged free surface shape. Busciglio et al. [28] also investigated the oscillation characteristics of free surface in an unbaffled stirred tank. The oscillation frequency was found to be different from the impeller rotational frequency, and was dependent on the vessel geometry and impeller rotation speed. The free surface movement has been also carefully investigated. Jahoda et al. [29] studied the surface oscillation in a baffled stirred tank through a numerical simulation. The simulation used a LES scheme for the turbulent modeling and a VOF model for the interface capturing. They concluded that the surface oscillation is caused by a macro-vortex located near the free surface. Kulkarni et al. [30] investigated gas entrainment through the VOF-LES scheme. According to their results, small cavity formed at the free surface is a source of gas entrainment. Fluid strain rate in the gas phase becomes stronger when the surface aeration occurs. Yang et al. [31] numerically investigated the free surface dynamics in an eccentrically stirred tank. As Yang et al. [31] discussed, the turbulent flow and free surface movement in an eccentrically stirred vessel are different from those in an on-axis stirred vessel. Nishikawa et al. [32] were the first who reported this finding. An eccentrically located impeller can reduce the mixing time, and the location of the surface vortex is changed. Hall et al. [33,34] compared an eccentric stirring in an unbaffled vessel with a centered stirring in a baffled vessel concerning the mixing time and the distribution of turbulent kinetic energy, which is one of the indicators of assessing the mixing efficiency. They indicated that the turbulent kinetic energy spreads widely in the eccentric stirring, that improves the mixing efficiency. Nomura and Iguchi [35] made experiments using an eccentric impeller to observe the shape of the surface vortex. They found that the developed surface vortex becomes inclined, and many low-density particles can be entrained from the surface vortex. Ohmi et al. [36] classified the shape of the surface vortex in an unbaffled cylindrical vessel stirred by an eccentric impeller. Galletti et al. [37] investigated the frequency of flow velocity oscillations in an unbaffled vessel agitated by an eccentrically located impeller. It has been found that the macro-instabilities is the main cause of these oscillations, and the Strouhal number can describe the variation of the oscillation frequency. Galletti et al. [38] also investigated the angle of the surface vortex inclination and the mixing time using a de-colorization experiment in an unbaffled vessel stirred by an eccentric impeller. Liu et al. [39] investigated bubble distribution in an eccentrically stirred unbaffled vessel with gas injection. The surface vortex shape and flow characteristics were found to be quite different from those in an on-axis stirred vessel. As discussed above, the dynamic behavior of the free surface during both on-axis and eccentric stirrings have been not clarified properly especially for the transient movement of the free surface. To shed light on the three-dimensional behavior of the free surface, we investigated the mechanisms of free surface deformation and vortex formation by conducting experiments and numerical simulation in an unbaffled vessel stirred by an on-axis or eccentric impeller. The main emphasis was placed on the transient structure of bubbles entrained from the turbulent free surface, and relationships between the dynamic behavior of the free surface and the flow structures. 2. Experimental procedures The configuration of the experimental system is shown in Fig. 1. An unbaffled cylindrical vessel was filled with water agitated by a paddle impeller with four thick blades. The blade tip-to-tip diameter d was 60 mm, the initial water level h was 320 mm, the inner diameter, D and height, H of cylindrical vessel was 192 and 400 mm (d = 0.3125 D; h = 1.667 D; H = 2.08 D), respectively. The distance between the initial water level h and the upper surface of blades was 40 mm. The clearance between the impeller midplane and the bottom wall, C was 274 mm (C = 1.427 D). The shaft diameter, ds, the impeller height, hI, 26
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lamp. The free surface shape was precisely measured by the following way. The halogen lamp was switched on to irradiate strong backlight, which was scattered and/or attenuated at the free surface. Pictures taken by the camera were blacked out at the location of the free surface. To obtain the oscillation amplitude and time-averaged profile of the free surface, the images were processed as shown in Fig. 3. First, the taken pictures were time-averaged for 2 s (1000 photos). Second, the difference in images between the time-averaged pictures and instant pictures was evaluated for each picture. Third, thus obtained picture was time-averaged for 2 s. Finally, the color of the time-averaged picture was inverted to better observe the picture details, as shown in Fig. 3. However, the free surface shape obtained by this procedure is assumed to over-estimate the free surface deformation because the backlight is scattered and/or attenuated at the projected area. Additionally, the transient free surface shape was obtained in a different technique. Strong light was irradiated from the upper side of stirring vessel to highlight the transient free surface shape. 3. Numerical methods and procedures 3.1. Governing equations for a liquid-gas flow The governing equations are the following balance equations of momentum and mass:
∂ (ρu ) + ∇ ·(ρuu ) = −∇p + ∇ ·μ (∇u + ∇uT ) + ρg + Fσ ∂t
(1)
∂ρ + ∇ ·(ρu ) = 0 ∂t
(2)
where ρ is the fluid density, u is the flow velocity, t is time, p is pressure, μis the viscosity of the fluid, and Fσ is the surface tension term. The present study adopted the Volume of Fluid (VOF) method (e.g. [40–42]) to capture the liquid-gas interface. The VOF method has many modifications to calculate the dynamics of free surface accurately and/ or easily. Among them, an algebraic VOF method has the advantage due to convenience to simulate multiphase flows in complicated geometry. Therefore, the algebraic VOF method was used in the present study. The algebraic VOF method developed by Weller [40] is expressed as:
Fig. 1. A schematic representation of unbaffled stirring vessel used in the present study.
and the impeller width, wI was 24 mm, 12 mm, and 6 mm, respectively. The impeller was rotated at constant rates of 300 rpm and 400 rpm. The distance between the shaft axis line and the vessel axis line was 0 and 60 mm. Hereinafter, this distance will be termed “eccentric length”. The driven flow was characterized by two non-dimensional parameters, Reynolds number (Re = nd2/ν) and Froude number (Fr = n2d/g), where n is the impeller rotation speed [rps], ν is the kinematic viscosity [m2s−1], g is the gravitational acceleration [ms−2]. The Reynolds number and Froude number are equal to 18,000 and 0.153 for 300 rpm rotation speed, and 24,000 and 0.272 for 400 rpm rotation speed, respectively. The free surface shape was recorded by a high-speed camera as shown in Fig. 2. The camera was fixed at the opposite side of a halogen
∂α + ∇ ·(αu ) + ∇ ·((1 − α ) αur ) = 0 ∂t
(3)
where α is the volume fraction of liquid and ur is the relative velocity between the liquid and gas phases. The definition of volume fraction α is:
α=
Liquid 1 0 Gas ⎨0 < α < 1 Interface ⎩ ⎧
(4)
In the present study, the contour of α = 0.5 depicted the free surface. The physical properties are updated according to the volume fraction α as:
ρ = ρl α + ρg (1 − α )
(5)
μ = μl α + μg (1 − α )
(6)
The relative velocity ur is treated as the compressive velocity. The third term on the left-hand side in Eq. (3) is discretized as:
∫Ω ∇·((1 − α ) αur ) dΩ
= ∑ ((1 − α ) αur )·Sf = ∑ ((1 − α ) αϕr )f
(7)
where Sf is the grid cell surface vector, ϕr is the volumetric compressive flux, and the subscript f is the cell face. The volumetric compressive flux ϕr is expressed as:
Fig. 2. Schematic representation of shooting method to obtain the free surface shape. 27
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Fig. 3. Procedure of image analysis: (a) an instant photograph, (b) a time-averaged picture, (c) a subtracted picture, (d) a time-averaged subtracted picture, and (e) a color-inverted picture.
where σ is the surface tension and k is the curvature of liquid-gas interface. Readers interested in more details on curvature calculation are referred to Rusche [41] and Yamamoto et al. [42].
the computational cost. The time step for the advection equation of liquid fraction was split and solved in the split time step. The number of sub-cycle iteration is 5. The temporal derivatives were discretized by a second order backward scheme except for Eq. (3). The multi-dimensional limiter for the explicit solution (MULES) scheme (e.g. [41]) was adopted for the temporal discretization of Eq. (3). The spatial discretizations are the second order linear interpolation except for Eq. (3). One of the TVD schemes, namely the second order linear-upwind scheme was used to avoid the numerical oscillations which can occur because the volume fraction in Eq. (3) has a steep gradient at the liquid-gas interface. The velocity and pressure fields were implicitly coupled by the PIMPLE algorithm, which is an improved version of PISO algorithm [44]. The OpenFOAM version utilized is 2.3.x.
3.2. Numerical procedures
3.3. Numerical grid resolution
The turbulence model used is the Large Eddy Simulation (LES) scheme. The solved equations in LES model are changed into
The numerical grid resolution was determined based on the Taylors micro-scale in the same way as Kulkarni and Patwardhan did [30]. They suggested that the Taylor micro-scale in a vessel agitated by an impeller with paddle blades is approximately 1–2 mm. Therefore, a numerical grid, the size of which near the impeller and the free surface is smaller than 1 mm, was used in the present study. Also, the Taylor micro-scale was estimated. In the present study, the turbulent energy dissipation rate ∊ was calculated as:
|ϕf | ⎤ ⎞ ⎛ |ϕf | n (ϕr )f = min ⎜Cα , max ⎡ ⎢ ⎥⎟ f |Sf | ⎣ |Sf | ⎦ ⎠ ⎝
(8)
where C α is the compression adjusting parameter. This parameter can change the degree of surface compression. The present study used unity for the compression adjusting parameter. The related numerical models are described in details in the relevant literature [40–42]. The surface tension force Fσ in Eq. (1) is set using the continuum surface force (CSF) model:
Fσ = σk∇α
(9)
∂ (ρτij ) ∂p¯ ∂ (ρu¯i ) ∂ ⎛ ∂u¯i ⎞ ∂ (ρu¯i u¯j ) = − + − + ⎜μ ⎟ + Fi ∂x j ∂x j ⎝ ∂x j ⎠ ∂x i ∂t ∂x j
(10)
∂ (ρu¯i ) ∂ρ =0 + ∂x i ∂t
(11)
where τ is the sub-grid scale stress tensor and the overlines indicate the filtered variables. The Sub-Grid Scale (SGS) turbulent kinetic energy is calculated on the basis of the transport equation of turbulent kinetic energy. The chosen cut-off filter is determined by the cubic root of grid cell volume (cubeRootVol). One-k-equation eddy viscosity model (oneEqEddy) is adopted for the sub-grid scale viscosity. The sliding grid technique directly expresses the rotation of the impeller. An open source software, OpenFOAM [43], was used for the present simulation. The sliding technique with the algebraic VOF method was simulated by interDymFoam, which is a solver implemented in OpenFOAM. The solver can handle the adjustable time step technique to reduce the calculation time. The time step was controlled using the local Maximum Courant number. In the present research, the Maximum Courant number was set to be 0.2. To capture the interface shape adequately, the solver adopted the sub-cycle algorithm for the advection equation of liquid fraction. When one utilizes this algorithm, the temporal resolution of free surface shape becomes high with the slight increase in
∊ = 2νSGS S¯ij S¯ij
(12)
where νSGS is the sub-grid-scale (SGS) viscosity, and Sij is the rate-ofstrain tensor. This calculation method is the same as that in the study of Murthy and Joshi [45]. The maximum turbulent energy dissipation rates are around 2–3 m2s−3 for 300 rpm rotation and around 5 m2s−3 for 300 rpm rotation. The Taylor micro-scale λ is expressed as:
λ2 =
15νu2 ∊
(13)
Finally, the Taylor micro-scale was estimated to be 2–3 mm for both the 300 rpm and 400 rpm conditions. Therefore, our calculation can resolve the turbulent vortices, the size of which is larger than the Taylor micro-scale. The vortices smaller than the Taylor micro-scale are modeled through the filtering used in a LES scheme. Finally, the total number of numerical grid cells was about 4,000,000. This suggests that the grid resolution used is sufficient to investigate the bulk and free 28
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surface flow dynamics through the LES scheme.
shape of the free surface is expressed as the time-averaged volume fraction α:
3.4. Boundary conditions
α¯ = At the sidewall and impeller surface, no-slip condition for velocity and zero gradient condition for volume fraction were imposed. At the top boundary, the flow velocity was determined by pressure distribution (pressureInletOutletVelocity), and the volume fraction was changed depending on the flow direction (inletOutlet). When the flow direction was toward the inner side of the numerical domain, the liquid fraction was set to zero. On the other hand, the boundary condition for liquid fraction was zero gradient to the boundary. Along the sliding grid interface, the cyclic arbitrary mesh interface (AMI) boundary condition was adopted to link the rotating and stationary boundaries. The residual error of algebraic matrix solver was set to be 1 × 10−6. The computation was carried out using the super-computing system in Kyoto University. The machine used was Cray CS400 2820XT, equipped with Intel Xeon Broadwell processors. It took about 4 weeks to complete simulation for a 15 s stirring by using 144 processors.
∫0
t'
αdt
(14)
where the t′ is the time-averaging interval, which is set to 5 s in the present study. The deviated free surface shape is expressed as the rootmean-square (RMS) value of volume fraction α:
α rms =
(α − α¯)2
(15)
The free surface deformation is increased with impeller rotation speed as seen in Fig. 4. This phenomenon is obvious because the turbulent intensity is enhanced with an increase of the impeller rotation speed. It is seen that the deformation magnitudes of simulated free surface shape are slightly smaller than those observed in the experiments. The possible reason for this difference is originated from the measurement method and calculation condition. As discussed in Section 2, the experimentally observed free surface corresponds to the projected area irradiated by light. The projected area involves the threedimensional free surface shape, which can be a source of the over-estimated deformation of the free surface. The calculation condition also can cause this difference. Time-averaging the free surface shape was started in 10 s after beginning the agitation. As Deshpande et al. (2017) [14] showed, the power number is converged to a specific value after about 20 impeller revolutions, which correspond to a time of 6–7 s in the present study. It was confirmed that the power number approaches a certain value within 10 s, although the surface vortex might not be fully developed yet. The depth of not fully developed vortex becomes smaller and the free surface is inclined at a lower angle compared to that at the fully developed vortex. Thus, insufficient number of impeller revolutions might cause this discrepancy. However, since this discrepancy is small, agreement between the experimental and numerical free surface deformations can be considered quite satisfactory. Finally, the experimental and numerically predicted transient free surface shapes were also compared. Fig. 5 shows the snapshots of surface vortex for the case of on-axis agitation at 400 rpm. Fig. 6 shows the time variation of simulated free surface shape near the on-axis impeller at the same rotation speed. In the both cases, small bubbles are entrained from the free surface following a number of steps: (1) The local surface vortex is elongated toward the impeller blade; (2) the local surface vortex reaches the rear surface of blade; (3) small bubbles are detached from the free surface; (4) the bubbles are moved on to the rear surface of blade; (5) the free surface is forced to move up by the next
4. Results and discussion 4.1. Comparison between experimental and numerical simulation results The main emphasis of present study was placed on the surface vortex formation and free surface deformation. Therefore, validation of the numerical model was carried out by comparing the free surface shapes obtained experimentally and numerically. First, the threshold of bubble entrainment was investigated depending on the impeller rotation speed in both the experiments and numerical simulation. In the experiments, the rotation speed was increased in an increment of 10 rpm. The experiments show that the critical rotation speeds of bubble entrainment for the on-axis and eccentric impellers are 390 rpm and 460 rpm, respectively. In the numerical simulation, air is entrained only in the case of the on-axis impeller at 400 rpm, while there is no air entrainment in the cases of the on-axis impeller at 300 rpm and the eccentric impeller at 300 rpm and 400 rpm. Therefore, the experimental and numerical conditions of bubble entrainment are in qualitatively agreement with each other. Next, the experimental and numerical free surface shapes and their oscillation magnitudes were compared. Deviation of the free surface shape from the time-averaged shape is shown in Fig. 4 for the stirring by on-axis impellers at 300 and 400 rpm. The calculated time-averaged
Fig. 4. The free surface shapes deviated from the time-averaged shapes when the liquid bath was stirred by the on-axis impellers of 300 and 400 rpm: (a) Experimental results, and (b) simulated results. 29
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Fig. 5. Experimentally obtained snapshots of surface vortex in the case of on-axis agitation of 400 rpm rotation.
Fig. 6. Time variation of simulated free surface shape near the on-axis impeller of 400 rpm rotation. 30
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Fig. 7. Side views of time-averaged free surface shape: (a) On-axis agitation of 300 rpm, (b) on-axis agitation of 400 rpm, (c) eccentric agitation of 300 rpm, and (d) eccentric agitation of 400 rpm.
approaching blade. These five steps are repeated. The similar time variation of the free surface movement could be observed in the numerical results as seen in Fig. 6. These findings suggest that the transient free surface shapes obtained by experiments and numerical simulation are in good agreement with each other. Thus, the above consideration confirms the validity and accuracy of the numerical model in terms of both the time-averaged and transient structures of the free surface. Therefore, the flow mechanisms and relationship between the free surface deformation and flow structures will be discussed on the basis of the simulated results in the following sections.
stirring conditions. The free surface deflection occurs at the point C in Fig. 8(c) and (d). A downward flow near the free surface causes this phenomenon. The liquid lifted by the upward flow at the narrow space must flow down somewhere to satisfy the conservation of mass. The downward flow is generated immediately after the fluid passes through the narrow space as shown in Fig. 9(b) and (d). 4.3. Transient flow characteristics 4.3.1. Local surface vortex As shown in Figs. 5 and 6, the local surface vortex is generated at the free surface. The generation mechanism of this vortex is discussed in this subsection. Fig. 10 depicts the free surface shape and pressure distribution near the impeller blades. The pressure is high in front of the impeller blades and low behind them. The low-pressure zone near the rear surfaces of blades forces the free surface to shift into a deeper place. Also, the high-pressure zone in front of the next approaching blade pushes the free surface up. Therefore, these results suggest that the pressure distribution around the rotating impeller is responsible for the local surface deflection.
4.2. Time-averaged flow characteristics First, the time-averaged free surface shapes formed in the on-axis and eccentric agitations were compared. Fig. 7 shows side views of the time-averaged free surface shape. The free surface was numerically defined as a surface where the contour of the time-averaged volume fraction equals to 0.5. The interface deformation becomes large with increasing the impeller rotation speed. And this effect is greater for the on-axis agitation than that for the eccentric agitation. The time-averaged free surface shapes are axisymmetric in the former case and nonaxisymmetric in the latter case. Fig. 8 shows top views of the timeaveraged free surface shape. In the case of the eccentric impeller, the free surface is elevated at the narrow zone between the impeller and the sidewall. To clarify the mechanism of these phenomena, the timeaveraged flow pattern was investigated. Fig. 9 shows the cross-sectional velocity vectors produced by the eccentric impellers. The white line indicates the cross-sectional free surface in Fig. 9. The discharged liquid flows and collides with the sidewall forcing the flow to change its direction into upward one. This upward flow elevates the free surface at the narrow space between the impeller and sidewall. As a result, the elevated part of the free surface (hereinafter referred to as “elevation”) located at the same distance from the sidewall as seen in Fig. 8(c) and (d). Also, this upward flow is assumed to decrease the free surface deformation making the air entrainment from the free surface difficult compared with the on-axis
4.3.2. Entire rotation of the free surface elevations The transient deformation of the free surface has been also investigated. The time variation of free surface shape agitated by the onaxis impeller at 400 rpm is shown in Fig. 11. The free surface elevations are located at two places and slowly rotated with a speed differing from that of impeller rotation. The rotation of impeller is 4/3 turn for 0.2 s, while that of surface elevations is about 2/5 for the same time interval. Besides, the number of free surface elevations is different from that of impeller blades which equals 4. It means that there is a complex relationship between the behavior of these elevations, the impeller geometry and rotation speed. Therefore, the fluid velocity near the impeller was considered more carefully. The cross-sectional velocity vectors are shown in Fig. 12. The large vortex is generated above the discharged flow, which does not develop toward the side wall straightly because of the turbulent oscillations. This oscillating discharged flow produces the large vortex oscillations, which affect the free surface movement. The 31
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Fig. 8. Top views of time-averaged free surface shape: (a) On-axis agitation of 300 rpm, (b) on-axis agitation of 400 rpm, (c) eccentric agitation of 300 rpm, and (d) eccentric agitation of 400 rpm.
Fig. 9. Time-averaged cross-sectional velocity vectors produced by the eccentric impellers: (a) 300 rpm, viewed from point A, (b) 300 rpm, viewed from point B and (c) 400 rpm, viewed from point A, and (d) 400 rpm, viewed from point B (Points A and B are shown in Fig. 8). 32
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Fig. 10. Time variation of free surface shape and pressure distribution near the on-axis impeller at a rotation speed of 400 rpm.
different from the on-axis stirring case. It means that the macro-instabilities do not occur at all or occurs in a small extent when the impeller is installed eccentrically. In this case, the free surface movement is characterized by small lifting and random deflection. The mechanism of these phenomena is explained below. The free surface behavior of eccentric 300 rpm agitation is similar. The rotation speed affects the magnitude of surface deflection and elevation. In the case of low-speed impeller rotation, the amplitude of free surface oscillation becomes smaller.
slow oscillation of this large vortex is generally referred as to macroinstabilities (MIs). Therefore, the macro-instabilities of large vortex is assumed to cause the rotation of two free surface elevations as shown in Fig. 12. In the case of the on-axis impeller rotating at 300 rpm, the macro-instabilities and related surface elevations are also generated. To characterize the macro-instabilities, the frequency of macro-instabilities per rotation speed (f/N) has been investigated in the previous studies. In the present study, the f/N value of free surface was around 0.22. In the study of Galletti et al., [38], the f/N value was about 0.17 with an on-axis Ruston turbine impeller. The f/N values are in good agreement with each other. This result also indicates that the entire rotation of free surface elevation is caused by the macro-instabilities. Fig. 13 shows the time variations of free surface shape when the bath is agitated by the eccentric impeller at 400 rpm. In this case, observations showed that the characteristics of free surface movement are
4.3.3. Trailing vortices Next, the structure of trailing vortices developed from the rear surfaces of blades was investigated. Fig. 14 indicates the trailing vortices in both the cases of on-axis and eccentric stirring. The vortex structure is depicted using contours of the second invariant of rate-of-
Fig. 11. Time variations of free surface shape of the bath agitated by the on-axis impeller of 400 rpm. 33
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Fig. 12. Transient cross-sectional velocity vectors near the blades in the case of on-axis impeller of 400 rpm: Broken line indicates the large vortex developed above the strong discharge flow.
As seen from the figure, the trailing vortices are elongated in the backward direction. The trailing vortices are thin, and they move slightly upward. Generally, counter-rotating vortex pairs are generated from the rear faces of paddle blades, and the size and shape of vortices are similar. However, in the present study, the blue-colored trailing vortices, generated at the upper half of rear blade, are much longer and stronger compared to red-colored vortices, generated at the lower half of read blade. Such a structure of trailing vortices is assumed to be a
strain tensor [46,47], which is expressed as:
ω¯ 2 ⎞ 1 Q = − ⎛S¯ij S¯ij − 2⎝ 2 ⎠ ⎜
⎟
(16)
where Q is the second invariant of rate-of-strain tensor, which is called Q-criterion, ω is the vorticity vector. The color of the vortex in this figure indicates rotating direction depicted by the circumferential component of vorticity. The depicted value of Q is 5000 s−2.
Fig. 13. Time variations of free surface shape in the water bath agitated by the eccentric impeller at 400 rpm. 34
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low-pressure zone generated behind the impeller blades, (2) entire rotation of the free surface due to the macro-instabilities, and (3) local free surface elevation and deflection at a location between the shaft and side wall due to the trailing vortices. The stirring method (on-axis or eccentric) determines the contribution of each of these phenomena to the surface deformation. Moreover, both the entire rotation of freesurface elevation and the local surface vortex formation affect the bubble entrainment. The bubbles are assumed to be entrapped in the following three steps: (1) The entire rotation of free surface elevations occurs due to the macro-instabilities. (2) The distance between the free surface and impeller blade becomes smaller because the free surface deflection is enhanced near the impeller shaft due to the entire rotation of free surface. (3) The bubble entrainment occurs because the free surface is pulled into the low-pressure zone behind the impeller blade generated by the impeller rotation. 5. Conclusion In the present study, the mechanism of surface vortex formation and free surface deformation in an unbaffled cylindrical vessel stirred by an on-axis or eccentric impeller was investigated. The experimental observations and numerical simulation were performed to clarify these phenomena. It is shown that the results of experiments and simulation are in good agreement with each other. The results of the present study can be summarized as follows:
• The deformation of free surface is small in the case of eccentric • • • •
Fig. 14. Structure of trailing vortices developed near the impeller rotating at 300 rpm and 400 rpm viewed from the side and the top: (a) on-axis stirring, and (b) eccentric stirring.
cause of the discharged flow, which is developed slightly upward due to the pressure distribution. The pressure between the impeller top and the free surface becomes lower because of the strong discharge flow and existence of the free surface. This low-pressure zone causes the discharged fluid to flow slightly upward. As a result, the trailing vortices reach the free surface and affect its shape. Therefore, the trailing vortices are assumed to be responsible for the free surface lifting and deflection at an intermediate distance between the side wall and shaft as seen from Fig. 11. In the case of eccentric stirring, the trailing vortices have short and complex geometry. They are rolled in the narrow gap between the impeller and the side wall. The structures of trailing vortices are changed and shortened at the narrow gap because the flow becomes complicated. The discharged flow transports the vortices forcing them to reach the free surface. Therefore, such vortices produce small and complicated elevation and deflection of the free surface. This is quite different from the large pseudo-periodic flow structure (macro-instabilities), which causes the slow rotation of surface elevations and deflections in the case of the on-axis stirring. The time-averaged free surface deformation has been discussed in terms of Froude number (Nagata [7], Rieger et al. [8], Deshpande et al. [14]). Froude number denotes the balance of inertial and gravitational forces which are significant for the free surface deformation. On the other hand, many factors affect the transient free surface deformation as discussed above. The following three phenomena are responsible for the surface deformation. (1) Local surface vortex formation due to a
•
stirring due to the upward flow developed at the narrow space between the impeller and sidewall. Three types of transient free surface deformation are observed: (1) local surface vortex, (2) entire rotation of free-surface elevation, and (3) local elevation and deflection. The low-pressure zone generated behind of impeller blades is responsible to the formation of local surface vortex. The entire rotation of free-surface elevations is caused by the macroinstabilities. In the case of eccentric stirring, the entire rotation of free-surface elevations occurs to a much smaller extent compared to the on-axis stirring case. The trailing vortices cause the local free-surface elevation and deflection.
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