A simulation of an unbaffled stirred tank reactor driven by a magnetic stirring rod was carried out in a moving reference frame. The free surface of unbaffled stirred tank was captured by Euler–Euler model coupled with the volume of fluid (VOF) method. The re-normalization group (RNG) kɛ model, large eddy simulation (LES) model and detached eddy simulation (DES) model were evaluated for simulating the flow field in the stirred tank. All turbulence models can reproduce the tangential velocity in an unbaffled stirred tank with a rotational speed of 150 rpm, 250 rpm and 400 rpm, respectively. Radial velocity is underpredicted by the three models. LES model and RNG kɛ model predict the better tangential velocity and axial velocity, respectively. RNG kɛ model is recommended for the simulation of the flow in an unbaffled stirred tank with magnetic rod due to its computational effort.

INTRODUCTION

Unbaffled stirred reactors are widely used in wastewater treatment, chemical and other process industries. Its feature lies in the strong tangential liquid motion. A central vortex on the liquid free surface can be formed with the increase of liquid level along the wall of the reactor and the trapping of air into the central vortex. The flow is typical of two-phase characteristics. Moreover, turbulence is usually predominant in unbaffled stirred reactors since it can offer excellent heat and mass transfer and hence enhance chemical reaction.

Numerical simulation on unbaffled stirred reactors can offer information about the water–air interface and local mixing. The volume of fluid (VOF) method is widely used to capture the water–air interface (Li et al. 1999; Ginzburg & Wittum 2001; Gao et al. 2003; Kim & Lee 2003; Mahmud et al. 2009; Haque et al. 2011). It determines the interface position by tracking the volume fraction of air or water. Zero or unity of air volume fraction means that this position is fully occupied by water or air. The local mixing is strongly associated with turbulence. Therefore, it is critical to choose appropriate turbulence model for simulating water–air two-phase flow in unbaffled stirred reactors. A lot of studies have been conducted to assess the merits of different turbulence models. The kɛ turbulence model was thought to be capable of offering accurate mean flow fields when used to simulate the two-phase flow (Armenante et al. 1997; Montante et al. 2006; Mahmud et al. 2009; Zhao & Guo 2010; Guo et al. 2012). However, Ciofalo et al. (1996) found that satisfactory results can only be obtained through the use of Reynolds stress model in simulating three dimensional (3D) water–air two-phase flow within unbaffled tanks. Mahmud et al. (2009) also concluded that Reynolds stress model can obtain a better result than the eddy-viscosity kɛ turbulence model. Although the use of Reynolds stress model can obtain a good result, it should be noted that a large amount of computational expense is required. Large eddy simulation (LES) is another powerful tool to simulate the turbulence in unbaffled stirred tanks (Alcamo et al. 2005; Lamarque et al. 2010). The numerical profiles of mean velocities were found to agree well with the measured data. Yang et al. (2012) conducted a numerical simulation of the liquid mixing in stirred tanks by using detached eddy simulation (DES). The computational data were in good agreement with the experimental data obtained by HU et al. (2010), showing that DES can be used to study the unsteady mixing characteristics in stirred tanks.

The aforementioned studies concern conventional impellers, either turbines or pitched-blade stirrers. Magnetic driven stirred tanks are also widely used in material and drug synthesis, food production, chemical products, etc. (Reichert et al. 2004; Hristov 2009; Rakoczy & Masiuk 2011; Rakoczy et al. 2014). It is likely to introduce a complex flow under the effect of magnetic force and further the significant influence on heat and mass transfer (Hristov 2009). Because of the absence of baffle, there is strong turbulence anisotropy in the tank. LES model could be a very promising method to obtain the hydrodynamic characters in an unbaffled stirred reactor driven by magnetic rod (Bertrand et al. 2012a). This study is to investigate the turbulent free-surface flow in an unbaffled stirred tank driven by a magnetic stirring rod using re-normalization group (RNG) kɛ model, LES model and DES model, respectively. The transient results are time-averaged over a long time to obtain the quasi-steady results. Our investigation is to identify a suitable turbulence model for predicting gas–liquid two-phase flow in an agitated tank.

GOVERNING EQUATIONS

Free-surface model

For an unbaffled stirred tank driven by a magnetic stirring rod, the water level rises along the tank wall while the air is trapped into the center of the tank. An immiscible interface exists between the water and the air. The VOF method is used to capture this immiscible interface by solving a single set of momentum equations and tracking the volume fraction of each fluid throughout the domain. Taking into account the stirred speed, both water and air are seen as incompressible fluid. The momentum equation of the VOF method is shown below. 
formula
1
where p is the mean pressure; gi is the acceleration due to gravity; Fi is the body force arising from the centrifugal and Coriolis terms; is the volume-weighted mixture density; is the volume-weighted mixture viscosity; and is the Reynolds stresses.
The tracking of the interface between the phases is accomplished by solving a continuity equation for the volume fraction of one phase. 
formula
2
where is the mass transfer from phase q to phase p and is the mass transfer from phase p to phase q.

Turbulence model

RNG kɛ model

The RNG kɛ turbulence model was derived from the instantaneous Navier–Stokes equations using RNG method. The transport equations for the RNG kɛ model are shown as follows: 
formula
3
 
formula
4
 
formula
5
where and are the inverse effective Prandtl numbers for k and , respectively. The values of the model constants and are taken to be 1.42 and 1.68, respectively, by default in ANSYS FLUENT.

Large eddy simulation

The governing equations for LES are derived by applying filtering operation to the Navier–Stokes equations. Thus, the large scale eddies are resolved directly by the filtered equations while the effect of the small-scale eddies must be modeled. The widely used LES model is the Smagorinsky model (Smagorinsky 1963), which reads as follows: 
formula
6
where is the filtered velocity and is the filtered shear stress tensor. The residual stress tensor σij is expressed as 
formula
7
The turbulent viscosity is modeled as follows: 
formula
8
where is the mixing length for subgrid scales and . In ANSYS FLUENT, is computed using 
formula
9
where is the von Karman constant, d is the distance to the closest wall, is the Smagorinsky constant and is the local grid scale. The value of the Smagorinsky constant is used by default in ANSYS FLUENT to be 0.1.

Detached eddy simulation

DES model is a blend of LES model and two-equation turbulence model. The LES model is used in the region far away from the wall, where large turbulence dominates. A two-equation turbulence model is used to bridge the gap between LES and the wall vicinity. In this study, shear-stress turbulence (SST) kω based DES model is adopted for DES simulation.

NUMERICAL SIMULATION

An unbaffled stirred tank with a magnetic stirring rod is modeled in the present study, as shown in Figure 1. The diameter (T) of the vessel is 0.15 m and the height is 0.3 m with the stationary liquid level (H) of 0.15 m. The diameter (D) of the stirring rod is 0.0216 m and the length (L) is 0.113 m. Owing to the rotation of the stirring rod, the computation must be carried out with respect to a 3D domain, as shown in Figure 2(a). The computational domain is discretized using an unstructured mesh consisting of 2.63 × 106 tetrahedral elements. The generated mesh is illustrated in Figure 2(b) with a finer grid arrangement near the rod.

Figure 1

Geometry of the unbaffled stirred tank.

Figure 1

Geometry of the unbaffled stirred tank.

Figure 2

Computational domain: (a) geometry, (b) grid.

Figure 2

Computational domain: (a) geometry, (b) grid.

The present simulation is carried out under the moving reference frame fixed on the stirring rod. The stirring rod is kept stationary within the moving reference frame. Meanwhile, the tank wall and the bottom of the tank are assigned an opposite angular velocity with the moving reference frame. The upper boundary is viewed as a symmetry plane. The VOF method was used to capture the water–air interface since the water level rises along the tank wall while the air is trapped in the center of the tank. Initially, the liquid level is 0.15 m while the air occupies the space from 0.15 m to the upper boundary. Both water and air are stationary at the initial moment.

The governing equations are solved by using the commercial software ANSYS FLUENT-13.0. The velocity–pressure coupling is handled by phase-SIMPLE algorithm. The second-order upwind scheme and bounded central differencing are used to discretize the convective terms for RNG kɛ model and LES, respectively. The Gauss-Green cell-based gradient approximation is used to evaluate the face value. Pressure is discretized by the PRESTO! scheme. To ensure the convergence, the computation is performed by using a transient solver with a time step of about 0.001 second. After 20 seconds, the computation tends to be stable. A quasi-steady flow field can be achieved by performing time-averaging with respect to unsteady results from 20 seconds to 50 s. A typical run time was 236 hours using RNG kɛ model, 389 hours using DES model and 440 hours using LES model. The time-averaged velocities are further averaged circumferentially for comparison with experimental data.

RESULTS AND DISCUSSION

Mahmud et al. (2009) measured the axial, radial and tangential velocities in a full-scale replica of an industrial reactor used for precipitation reactions using 1D fiber-optics. Measurements were taken at different heights and locations depending on the stirrer speed. The error for the mean velocity measurement in the majority of the reactor was approximately 1% of the impeller tip speed. The experimental details can be found in Mahmud et al. (2009). In the present study, the predicted data by three turbulence models are compared with the experimental data.

Mean velocity distribution

Mean tangential velocity

Figure 3 shows the comparison of experimental and simulated mean tangential velocity () distributions at various heights with a rotational speed of 150 rpm. In the range of , all turbulence models underpredict tangential velocity, but the predictions by LES turbulence model are relatively close to the experimental data. In the range of , higher tangential velocities are predicted by three turbulence models. In this case, the predicted data obtained by LES model provide a more realistic tangential velocity. All the predicted tangential velocities are in good agreement with the experimental data, especially LES.

Figure 3

Comparison of mean tangential velocity distribution at various heights for a rotational speed of 150 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─, DES model.

Figure 3

Comparison of mean tangential velocity distribution at various heights for a rotational speed of 150 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─, DES model.

As pointed out in the introduction, tangential motion dominates in the unbaffled stirred reactor. Since the predictions of LES model are closer to the experimental data, the contour of tangential velocity at the plane of XZ obtained by LES model is shown in Figure 4. A symmetric pattern can be observed for the tangential velocity. The tangential motion of water is weak near the axis of the reactor.

Figure 4

Contours of tangential-velocity at the plane of XZ obtained by LES model for a rotational speed of 150 rpm.

Figure 4

Contours of tangential-velocity at the plane of XZ obtained by LES model for a rotational speed of 150 rpm.

Figures 5 and 6 show the mean tangential velocity () distributions at different heights with rotational speeds of 250 rpm and 400 rpm, respectively. As shown in the figures, the predicted data have good agreement with the experimental data with increasing height. The trends show that the predicted data simulated by DES model have a better agreement than those obtained by LES model and RNG kɛ model when the stirring rod is at a high speed. For all rotation speeds, there is a discrepancy between the experimental data and the computed data near the wall. It may be ascribed by the near-wall turbulence under swirling condition since the standard wall function was designed based on the flow parallel to the wall.

Figure 5

Comparison of mean tangential velocity distribution at various heights for a rotational speed of 250 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─, DES model.

Figure 5

Comparison of mean tangential velocity distribution at various heights for a rotational speed of 250 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─, DES model.

Figure 6

Comparison of mean tangential velocity distribution at various heights for a rotational speed of 400 rpm. △, experimental; ─, RNG kɛ model; ─LES model; ─, DES model.

Figure 6

Comparison of mean tangential velocity distribution at various heights for a rotational speed of 400 rpm. △, experimental; ─, RNG kɛ model; ─LES model; ─, DES model.

Mean axial velocity

Figure 7 depicts the mean axial velocity () distributions at different heights with the rotational speed of 150 rpm. The overall trend is good although there exists a discrepancy near the axis. All turbulence models underestimate the axial velocities near the axis. In the range of , the predictions of axial velocity by LES model are evidently smaller than predictions by the RNG kɛ and DES models. Overall, the predictions of axial velocities using RNG kɛ model are better than LES and DES models.

Figure 7

Comparison of mean axial velocity distribution at various heights for a rotational speed of 150 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─, DES model.

Figure 7

Comparison of mean axial velocity distribution at various heights for a rotational speed of 150 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─, DES model.

Figure 8 depicts the axial velocity distribution at the plane of XZ obtained by RNG kɛ model at the rotational speed of 150 rpm. There exists a slightly asymmetry in the axial velocity. Moreover, the magnitude of axial velocity is smaller than that of the tangential velocity.

Figure 8

Contours of axial-velocity at the plane of X-Z obtained by RNG kɛ model for a rotational speed of 150 rpm.

Figure 8

Contours of axial-velocity at the plane of X-Z obtained by RNG kɛ model for a rotational speed of 150 rpm.

Figures 9 and 10 show the comparisons of the experimental and predicted mean axial velocity () at different heights with the rotational speeds of 250 rpm and 400 rpm, respectively. In contrast to the experiment of 150 rpm, a lot of air was trapped in the region adjacent to the axis of reactor for 250 and 400 rpm. Thus, no water velocity can be obtained for the region of r < 0.038 m at a height of 0.1 m for a rotational speed of 250 rpm and the same occurs in the region of r < 0.025 m at a height of 0.06 m for a rotational speed of 400 rpm. As shown in Figure 9, when the heights are 0.06 and 0.072 m, the predicted data and the experimental data agree fairly well, especially the predicted data obtained by DES model. As shown in Figure 10, in the range of , the predicted data depict that there exists a big difference among RNG kɛ, DES and LES models. RNG kɛ model can get a better relative result.

Figure 9

Comparison of mean axial velocity distribution at various heights for a rotational speed of 250 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─, DES model.

Figure 9

Comparison of mean axial velocity distribution at various heights for a rotational speed of 250 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─, DES model.

Figure 10

Comparison of mean axial velocity distribution at various heights for a rotational speed of 400 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─, DES model.

Figure 10

Comparison of mean axial velocity distribution at various heights for a rotational speed of 400 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─, DES model.

Mean radial velocity

Figure 11 shows the predicted mean radial velocity () distributions at various heights with a rotational speed of 150 rpm. The simulated radial velocities are very weak, which is in accordance with the radical velocity distribution, as shown in Figure 12. A big discrepancy between the experimental data and the computed data is observed in all simulations. A similar phenomenon was also observed by previous studies (Ciofalo et al. 1996; Brucato et al. 1998; Halász et al. 2007; Bertrand et al. 2012b; Busciglio et al. 2013). The reason may lie in that the radial and axial velocities are much smaller than tangential velocity at quasi-steady state. Thus, a small discrepancy in the prediction of tangential velocity may lead to a big error in the prediction of radial velocity because of the mass conservation. Another reason lies in the lack of information of the effect of strong swirling on turbulent stress, which has been a big challenge for computational fluid dynamics (CFD) simulation.

Figure 11

Comparison of mean radial velocity distribution at various heights for a rotational speed of 150 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─DES model.

Figure 11

Comparison of mean radial velocity distribution at various heights for a rotational speed of 150 rpm. △, experimental; ─, RNG kɛ model; ─, LES model; ─DES model.

Figure 12

Contours of radial-velocity distribution at the plane of XZ obtained by RNG kɛ model for a rotational speed of 150 rpm.

Figure 12

Contours of radial-velocity distribution at the plane of XZ obtained by RNG kɛ model for a rotational speed of 150 rpm.

Predictions of free surface

Figure 13 depicts the predicted free surface in the xz plane. The predicted free-surface shapes are similar to those shown in Figure 3 of Mahmud et al. (2009). The liquid levels at the center of the tank can be estimated free surfaces, which drop about 0.02 m, 0.06 m and 0.125 m for rotational speeds of 150 rpm, 250 rpm and 400 rpm, respectively. As the rotational speed increases, the height of the liquid level along the tank wall elevates with the depth of the vortex which is deformed. The vortex depth, i.e., the difference between the highest water level and the lowest water level, is used to characterize the vortex formation, which depends upon the stirrer diameter, stirrer speed, liquid height at rest, the physical properties of the liquid and tank geometry. It is expressed as a power function of rotation speed (Markopoulos & Kontogeorgaki 1995; Rao et al. 2009). The present computed vortex depths are 0.03 m, 0.08 m and 0.17 m for rotational speeds of 150 rpm, 250 rpm and 400 rpm, respectively, which is proportional to rotational speed.

Figure 13

Predicted free surface of the cross section XZ axis: (a) 150 rpm, (b) 250 rpm, (c) 400 rpm.

Figure 13

Predicted free surface of the cross section XZ axis: (a) 150 rpm, (b) 250 rpm, (c) 400 rpm.

CONCLUSIONS

In this work, the flow field in an unbaffled stirred tank driven by a magnetic stirring rod has been investigated in detail under moving reference frame. The VOF method is used to capture the free surface of the fluid. RNG kɛ model, LES and DES models are used to describe the turbulence effect. All turbulence models predict the similar flow fields with the rotation speeds of 150 rpm, 250 rpm and 400 rpm, respectively. The LES model behaves the best in the prediction of tangential velocity while the RNG kɛ model predicts the best result of axial velocity. There is the roughly same error in the prediction of radial velocity, which should be improved further. The predicted vortex depth is proportional to the rotation speed. The comparison of turbulence models proves that the RNG kɛ, LES and DES models can be used to simulate the water–air flow in an unbaffled stirred tank. However, taking into account computational effort, RNG kɛ model can be a good candidate.

ACKNOWLEDGEMENTS

This study was supported by Innovation Program of Shanghai Municipal Education Commission (no. 14ZZ136) and the Shanghai Committee of Science and Technology (no. 13ZR1427600).

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