CFD analysis of a rotor-stator mixer with viscous fluids

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Abstract

The characterization of the hydrodynamics of a rotor-stator mixing head has been carried out in the laminar regime with viscous Newtonian fluids. The rotor-stator considered is a very common design composed of a flat blade rotating in a fixed slotted cage. A numerical methodology has been used based on the virtual finite element method to model the velocity patterns, estimate the distribution of shear stress and the flow rate through the head. We have found that the numerical prediction of the power consumption and flow profiles compare well with experimental data. The generation of a pseudo-cavern around the mixing head and how it scales with the Reynolds number have also been investigated, showing that there is a minimum speed limit below which the rotor-stator cannot be used.

Introduction

Rotor-stators are a particular class of mixers commonly used in the petrochemical and cosmetic industries for the production of liquid dispersions and emulsions. When operated at high speed with low-viscosity fluids, they generate a highly turbulent flow field and the pumping discharge is sufficient to maintain the flow in the vessel. Rotor-stators can also be used with viscous fluids. In such a case, the flow around the mixing head is much more complex, exhibiting many of the features associated with non-Newtonian mixing in the laminar regime like segregations, caverns, etc. (Doucet et al., 2005). Although important from an industrial application standpoint, the knowledge of the hydrodynamics in this regime is very limited and deserves more work.

CFD (computational fluid dynamics) is often used to simulate fluid flow with moving parts like in mechanically stirred vessels. This approach provides a wealth of information that can help improving the design of the mixer and its operating efficiency. Detailed velocity, shear rate and flow rate information can also be obtained experimentally with advanced measuring techniques like particle image velocity (PIV) and laser-doppler anemometry (LDA) (Li et al., 2005), but it is essentially a research tool dedicated to small scale bench experiments.

The use of CFD for the flow simulation in a rotor-stator is not an easy task due to the complex topology that continuously changes with time. The virtual finite element method (VFEM), which allows computing flow problems with evolving boundaries without resorting to the re-meshing of the complete flow domain at each time step, will be used in this work. This method is a particular class of fictitious domain method (also called domain embedding method) and it is an advanced version of the Lagrange multiplier fictitious domain method mostly developed by Glowinski et al., 1999, Glowinski et al., 2000. With the VFEM, the moving parts are not represented as geometrical obstacles which position changes with time but rather by constraining the flow equations in a mathematical sense. The moving parts are in practice replaced by control points on which the velocities are considered as kinematics constraints enforced into the variational formulation of the equations of change by means of Lagrange multipliers. This method was specifically developed for the analysis of flow problems in enclosure containing internal moving bodies. Its application to mixing problems already includes the modeling of a conical helical mixer (Dubois et al., 1996), a helical ribbon mixer (Bertrand et al., 1997), planetary kneaders (Tanguy et al., 2001), twin-screw extruders (Bertrand et al., 2003) and turbines in centered and eccentric configurations (Rivera et al., 2004).

The numerical modeling literature on rotor-stator hydrodynamics is pretty scarce. Böhm et al. (1998) presented a 2D parallel multigrid finite volume solver to predict unsteady flow in a rotor-stator configuration using a moving-grid technique. They computed the transient phenomena of creation and destruction of eddies during the start-up of a rotor at a Reynolds of 1000. They also highlighted the presence of a secondary flow between the blades of the rotor. Calabrese et al. (2002) performed a 2D sliding mesh simulation of an in-line slotted rotor-stator mixer using the Reynolds average Navier–Stokes (RANS) equations with a kε turbulence model closure. He showed that for a turbulent flow at a Reynolds number of 104, the mechanical forces generated in the shear gap are not the main responsible for emulsification and dispersion processes. The jets and the swirls discharged from the slots provide the central mechanisms for achieving mixing. Thakur et al. (2002) performed a 3D sliding mesh finite volume simulation of a centrifugal blower using a quasi-steady rotor-stator and kε turbulence model. They showed that the prediction of both static pressure rise, horsepower and mass flow rate compared well with experimental data.

In the above studies, the sliding mesh method was used to bypass the excessive CPU time that would be associated with the generation of a grid at each time step. Nevertheless, several issues remain unresolved with the turbulent sliding mesh method in particular the local and global flux conservations on the interfaces (Thakur et al., 2002).

The objective of the present work is the characterization of the hydrodynamics inside and outside the vicinity of a rotor-stator mixing head in the laminar regime with viscous Newtonian fluids. In the forthcoming, we first summarize the experimental part of the study and the hydrodynamics characterization in terms of power draw and flow patterns. Then, we briefly introduce the two computational models used in this work and discuss the simulation results in terms of velocity patterns, shear stress, pseudo-cavern and flow rate through the stator at the light of the available experimental data.

Section snippets

Summary of the experimental investigation

A rotor-stator mixer immersed in a 17 L volume vessel was used. It was provided with a shaft mounted torque transducer and a speed encoder allowing to establish the power curve. The rotor-stator head manufactured by VMI-Rayneri (France) is shown in Fig. 1. It is based on a 4-blades impeller rotating in a fixed stator having 72 slotted orifices, 8 aspiration orifices on the upper side and a free opening at the bottom. Fluids tested included Newtonian and non-Newtonian inelastic fluids. The whole

CFD modeling

The flow of an incompressible fluid in a given domain Ω with boundary Γ is expressed by the classical Navier–Stokes equationsρVt+V.gradV+gradp+divτ=finΩ,divV=0inΩ,where V(Vx,Vy,Vz) is the velocity, ρ the fluid density, p the pressure and f a body force. To close the momentum equation (4), the stress tensor τ is expressed as a function of the velocity field as expressed by a rheological equation of stateτ=-2η(|γ˙|)γ˙,where γ˙=12[gradV+(gradV)T] is the rate-of-strain tensor. In this work the

Results and discussion

The first part of the work consisted of optimizing the finite element mesh size ensuring that the velocity gradients in the high-shear region (especially the rotor-stator gap) could be captured with a reasonable accuracy. Mesh partitioning was used for this purpose. The number of control points used to describe the rotor was also optimized after several preliminary computations that showed its sensitivity with respect to the simulation results (velocity, shear stress, pressure) and the CPU

Conclusions

In this study, 3D CFD was used to characterize the hydrodynamics of a rotor-stator immersed in a vessel in the laminar regime. The simulations were validated against experimental data on torque showing a very good agreement. The analysis of the stress field showed that the maximum stress is found in the clearance between the rotor blade tip and the stator. The presence of the pseudo-caverns evidenced in a previous experimental study was highlighted by analyzing the shear stress field and

Acknowledgements

The financial support of TOTAL S.A. is gratefully acknowledged.

References (14)

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