LES of Mixing Time in a Stirred Tank with a Rushton Turbine

LES of Mixing Time in a Stirred Tank with a Rushton Turbine

MIN Jian, GAO Zheng-ming and SHI Li-tian (闵健，高正明，施力田) School of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China

Abstract The mixing process in a stirred tank of 0.476 m diameter with a Rushton turbine was numerically simulated by using computational fluid dynamics (CFD) package FLUENT6.1. Large eddy simulations (LES) with Smagorinsky-Lilly subgrid scale model were computed to simulate mixing process, the turbulent flow field and mixing time. The impeller was modeled using the sliding mesh model. The hybrid meshes method was chosen to divide the whole calculated region into two parts, one compassed impeller using the unstructured grids, and the other was the bulk of tank using the structured grids. Based on the response curve of tracer, the mixing time predicted by LES was in a better agreement with the experimental than the traditional Reynolds-averaged Navier-Stokes (RANS) approach. The results show that LES is a good approach to investigate unsteady and quasi-periodic behavior of the turbulent flows in stirred tanks.

Keywords LES, Smagorinsky-Lilly subgrid scale model, mixing time, Rushton turbine

INTRODUCTION

The mixing time, θm, is the time required to mix the added secondary fluid with the contents of the vessel until a certain degree of uniformity is achieved, typically θ95 meant to reach 95% of the final concentration. Knowledge of mixing time is required for the optimum design of stirred tank/reactors. In the last two decades, progress has been achieved in the CFD simulation of mixing process owning to the great achievement in the computer technique. Ranade[1] used numerical simulations to give the detail on the flow and bulk mixing generated by a downflow-pitched blade turbine in a fully baffled cylindrical vessel. Lunden[2] simulated pulse experiments by solving the material balance of the tracer in three dimensional flow fields with Rushton turbine (DT6). They suggested that the quality of results was highly dependent on exact fluid dynamic computations, especially concerning turbulence modeling. Large eddy simulation (LES), first adopted in stirred tank by Eggles[3], was proved to be a good method to investigate turbulent flows, unsteady and quasi-periodic behavior. Subsequently, several authors, Revstedt[4] pointed out that LES could provide details of the flow field that cannot be obtained with so-called Reynolds averaged equations and corresponding models. The previous investigators all focused on studying the three-dimensional velocities and the turbulent kinetic energy in a stirred tank and also proved that the LES was a good approach to investigate turbulent flows in industrial application of practical importance.

In this work, the1LES with Smagorinsky-Lilly subgrid model was introduced firstly to simulate mixing process through monitoring the concentration of tracer to get mixing time in a stirred tank with DT6, and its results were compared with experimental.

1 Physical and Computational configuration

The stirred tank used in this work was a perspex vessel of 0.476 m in diameter with flat bottom and four baffles. The DT6 speed was 240 r?min-1, which corresponds to the Reynolds number of 1.4×105, i.e. the fluid flow was in the turbulent region. Details of the apparatus were shown in Fig. 1.

1闵健，1974，男，博士，流体混合与反应器工程

The θ95 was measured from changes in conductivity after the introduction of a small quantity of tracer, saturated potassium

chloride (KCl) solution. 10ml of the tracer was added to the free

surface of the liquid between two baffles. The detector was

always mounted at the same position near the bottom of the tank opposite to the adding point. The output from the conductivity

meter was taken through an analogue filter, an amplifier, an analogue to digital converter and then stored on a computer for subsequent analysis. Fig. 1. Cross-section of the tank (Unit: mm)In order to compare with the experimental results, the same condition was chosen in the simulation as in the experiment, whose detail position of adding and monitoring points was given in the Fig. 1. The position P 3 of monitor point in the simulation was chosen to compare with experimental, and some other points were chosen to monitor to get the details of whole tank.

2 MATHEMATICAL METHOD

2.1 The LES and subgrid-scale model

LES is situated somewhere between Direct Numerical Simulation and the Reynolds-averaged Navier-Stokes (RANS) approach. Basically large eddies are resolved directly in LES, while small eddies are modeled. The most basic of subgrid-scale models was the Smagorinsky-Lilly model, In this model, the eddy viscosity is modeled by:S L

S t

2ρμ

=, where L S is the mixing length for subgrid scales and ij ij S S S 2≡, ij S is the rate-of-strain tensor for the resolved scale. In Fluent, L S is computed using: L S ＝min （Kd ，C S V 1/3），K is the V on Karman constant, d is the distance to the closest wall, V is the volume of the computational cell, C S is set to 0.1 in this paper.

2.2 Hybrid Meshes

For the mixing tank simulations performed, the computational mesh is made up of two parts: an inner rotating cylindrical volume enclosing the turbine, and an outer, stationary volume containing the rest of the tank. Mesh strategy was adopted the technique of combination with the structure and unstructured grid. The part of the impeller divided by unstructured tetrahedral cells and was densified to get more accurate description of impeller. And the other, the bulk of the tank, was adopted the multi-block method, in which the hexahedral cells were used to decrease cost of calculation. The illustrations of the computational mesh were given in Fig. 2.

Fig. 2. Vertical (left), horizontal (middle) and impeller (right) view of mesh

2.3 LES procedure

The first step was to calculate continuity and velocity equations using the k -ε model . After the flow field is

somewhat converged, the result was used as an initial condition for LES. The simulation chose the second-order implicit formulation for temporal discretization and the central-differencing scheme for spatial. The transient impeller motion was modeled using the sliding mesh model, which is in fact a transient method. The step was to calculate continuity and velocity equations until the flow became statistically steady through monitoring the torque of impeller.

The second was that the equation for concentration of tracer was solved at different time, then to get the mixing time. The varying of concentration simulation with time is unsteady-state problem. The injection of tracer is assumed not to affect the flow. Therefore, the separation of moment and material balance equations is chosen to reduce the computational effort. Based on the physical coordinate point, the adding point of tracer in simulation was defined as the several grids near it to ensure that the volume of simulation was the same as that of experiment. The concentration of tracer was initialized as 1 in the adding region, and in the else region as 0. The LES was run on XEON dual processors (Pentium IV) machines (Dell) with 1 Gb of memory, 2 GHz clock frequency and LINUX operating system. The computation was done on four processors in parallel for flow field spending about 4 weeks, and for concentration field about 3 weeks.

3 RESULTS AND DISCUSSION

3.1 Simulation the field flow and concentration of trace in the whole tank

The velocity of field flow gotten through LES and the k-ε model[5] were drawn in Fig. 3. In a real stirred tank, there are large vortices and resulting macro-instabilities, which promote tracer mass exchange through this boundary. The standard k-ε model can not fully account for this phenomenon. But LES could catch the details of vortices even very small scale to Taylor and Kolmogorov micro scale[6] rested with the mesh size in simulation. So LES could give the more accurate result for the instantaneous flow than RANS. As can be seen from the Fig. 3, the boundary of axial-radial circulation loops in LES, comparing with that gotten from RANS, was broken and irregular, and in else bulk flow took on a lot of large and small vortices. The mass exchange among these vortices in the whole tank could be resolved accurately.

Fig. 3. Contour of Instantaneous Velocity Magnitude( left, RANS; right, LES) (m/s) N=240 r?min-1

Fig. 4. Contour of Dimensionless Concentration of KCl ( t=2s, 4s, 6s)

After the velocity and eddy diffusivity fields obtained, the mixing process can be computed by solving the conservation equation of the tracer. Fig. 4 shows the concentration distribution of tracer in the whole tank vs. time in LES. From the contour of concentration can be seen the mixing process directly.

3.2 The response curve of tracer and mixing time θ95

D i m e n s i o n l e s s c o n c e n t r a t i o n o f K C l

Time (s)

The response curves of tracer, three points simulations and experimental, were normalized to draw in Fig. 5. The mixing time was taken as the time at which the tracer concentration latest reached within 1±5％ of the final tracer concentration.

The θP3 in the simulation was chosen to compare with experimental. LES can not only get the correct mixing time, but also the reasonable response curve of tracer. Mixing time strongly depends on the positions for tracer adding and monitoring. As seen from Fig. 5., the mixing time of monitoring in tank bottom P 3 is close to that P 2 in impeller area, however it is shorter greatly than that in liquid surface P 1. And the relative error of the value P 3 compared with experimental is 3.4％.

Fig. 5. The Response Curve of KCl in LES and Experimental CONCLUSIONS

Large eddy simulations of the flow field and mixing time in a stirred tank of 0.476 m in diameter with a Rushton turbine have been performed. The unsteady and quasi-periodic behaviors of the turbulent flows in the stirred tank were well captured, which indicated that LES is a very good approach for simulating this kind of complex flow situations. The response curve of tracer and mixing time predicted by LES is in a good agreement with the experimental.

REFERENCES

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