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) *k*–*ɛ* 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 *k*–*ɛ* model predict the better tangential velocity and axial velocity, respectively. RNG *k*–*ɛ* 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 *k*–*ɛ* 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 *k*–*ɛ* 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) *k*–*ɛ* 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

*p*is the mean pressure;

*g*

_{i}is the acceleration due to gravity;

*F*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.

_{i}*q*to phase

*p*and is the mass transfer from phase

*p*to phase

*q*.

### Turbulence model

#### RNG *k*–*ɛ* model

*k*–

*ɛ*turbulence model was derived from the instantaneous Navier–Stokes equations using RNG method. The transport equations for the RNG

*k*–

*ɛ*model are shown as follows: 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

*σ*is expressed as The turbulent viscosity is modeled as follows: where is the mixing length for subgrid scales and . In ANSYS FLUENT, is computed using where is the von Karman constant,

_{ij}*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) *k*–*ω* 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 × 10^{6} tetrahedral elements. The generated mesh is illustrated in Figure 2(b) with a finer grid arrangement near the rod.

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 *k*–*ɛ* 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 *k*–*ɛ* 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.

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 *X*–*Z* 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.

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 *k*–*ɛ* 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.

#### 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 *k*–*ɛ* and DES models. Overall, the predictions of axial velocities using RNG *k*–*ɛ* model are better than LES and DES models.

Figure 8 depicts the axial velocity distribution at the plane of *X*–*Z* obtained by RNG *k*–*ɛ* 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.

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 *k*–*ɛ*, DES and LES models. RNG *k*–*ɛ* model can get a better relative result.

#### 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.

### Predictions of free surface

Figure 13 depicts the predicted free surface in the *x*–*z* 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.

## 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 *k*–*ɛ* 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 *k*–*ɛ* 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 *k*–*ɛ*, 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 *k*–*ɛ* 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).