A virtual finite element model for centered and eccentric mixer configurations
Introduction
In many industrial processes, tank-based mixing must be operated in the laminar regime due to the high viscosity of the phases to be mixed. The challenge is then to ensure a good mixing efficiency to offset for the lack of turbulence. Although it has been a classical subject in mixing science for many years (Harnby et al., 1992), the fluid mechanics of laminar mixing is knowing a renewed interest. A number of experimental investigations has been published in the case of stirred vessels with centered impellers (Tanguy et al., 1992) as well as with less conventional mixers based on non-symmetric impeller configurations (Alvarez et al., 2002, Ascanio et al., 2002), complex impeller arrangements as those found in planetary kneaders and coaxial mixers (Dubois et al., 1996, Tanguy et al., 1997, Tanguy et al., 2001 (Chapter 12); Espinosa-Solares, 1998).
In the same period, flow numerical simulations using Computational Fluid Dynamics (CFD) have gained full acceptance in the scientific mixing community as a tool to investigate mixing patterns in stirred vessels. Many studies focused on the hydrodynamic complexities generated by the rotating impeller at high viscosity or with rheologically complex fluids (Harvey & Rogers, 1996; Tanguy et al., 1996, Tanguy et al., 1998, Tanguy et al., 2001 (Chapter 12); Ranade, 1997, Harvey et al., 2000, Zalc et al., 2001, Bujalski et al., 2002). Mixing flow simulations were carried out based on the finite volume or finite element resolution of the classical equations of change in conjunction with either moving meshes (Demirdzic & Peric, 1990) or sliding meshes (Perng & Murthy, 1992) to depict the kinematics of the impeller relative to the vessel tank and internals. Another approach called the virtual finite element method (VFEM) was introduced by Bertrand et al. (1997). This method is a particular class of fictitious domain method (Glowinski et al., 1994) and was specifically developed for the analysis of flow problems in enclosures containing internal moving parts like with an agitated vessel. Briefly, the method consists of representing the rotating impeller by a series of control points (CP) located on its surface and to enforce the kinematics of these control points by constraints in the sense of constrained optimization. This method has been successfully used for modeling complex mixers such as planetary kneaders (Tanguy et al., 1996, Tanguy et al., 2001 (Chapter 12)), conical mixers (Dubois et al., 1996) and twin-screw extruders (Bertrand et al., 2003). A very good agreement was found when comparing the results obtained by VFEM with experimental data.
In mixer design, off-centered impellers are sometimes used to limit cavern formation in the absence of baffles (Alvarez et al., 2002). The flow simulation of eccentric mixers poses special numerical challenges. Indeed, as no symmetry conditions can be used to simplify the resolution, the unsteady version equations of change must be solved and a new computational mesh is a priori required at every time step, which is a major drawback in terms of computational efforts.
The purpose of the present study is to highlight the capabilities of the VFEM to model the fluid mechanics of mixing with eccentric impellers without resorting to remeshing. The centered impeller configuration will be first considered to allow the comparison with a conventional Galerkin finite element method (FEM). Special attention will be given to the effect of the meshing strategy and the influence of the number of CP on the computational accuracy. The method will then be applied for the simulation of the eccentric configuration.
Section snippets
Computational model
To numerically predict the three-dimensional flow pattern in a stirred tank, two formulations of the Navier–Stokes equations can be established depending whether the Eulerian viewpoint (Fig. 1a ) or the Lagrangian viewpoint (Fig. 1b) is used. In the Eulerian viewpoint (laboratory frame of reference), the impeller rotates in a fixed vessel. The flow equations are expressed as follows:where V (Vx, Vy, Vz) is the velocity vector,
Mesh and control point sensitivity analysis
Fig. 3 shows the mixing system used for the present investigation. It consists of an unbaffled stirred tank of 165 mm in diameter and a liquid height-to-diameter ratio of 1. The tank is equipped with a Rushton turbine having a diameter D of 65 mm attached to a rigid shaft, which can be horizontally displaced from the centerline of the tank by a distance x. The rotation speed ω is 100 RPM. The fluid used in the simulations has a dynamic viscosity μ of 50 Pa s and a density ρ of 1000 kg/m3. Based on
Flow induced by an eccentric impeller
The VFEM previously described and validated was used to simulate the flow generated by a single eccentric impeller located at (1/4)D in the radial direction and (2/3)H in the axial direction (Fig. 3). The mixer configuration involved a Rushton turbine rotating at ω = 100 rpm. A Newtonian fluid having a dynamic viscosity of 0.5 Pa s and a density of 1000 kg/m3 was considered. Such conditions correspond to a laminar mixing flow at a Reynolds number of 14.
The Eulerian viewpoint was used for the
Conclusion
The objective of this paper was to establish the optimal conditions for the use of the virtual finite element method in mixing problems. A centered mixer configuration and a non-trivial eccentric mixer configuration were used as test cases. The effect of the number of CP and meshing strategy (geometry partitioning and mesh sizing) was investigated. The policy of “maximum active CP with minimum CP per element” gave excellent results. It was found that the percentage of active CP is a key factor
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2006, Chemical Engineering ScienceCitation Excerpt :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). 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.