Modeling Turbulence in Stirred Vessels: A Review and Recent Developments

Abstract

Prediction of the mixing of multi-component fluids is important in many chemical process applications. Although laminar mixing is a complicated process per se (involving multi-component diffusion coefficients, for example), there is a far greater challenge in predicting mixing in turbulent flows because of their intrinsic, chaotic nature. In turbulent flows, large-scale eddies with coherent structures are mainly responsible for the mixing of passive scalars. The large-scale eddies embody themselves in the form of identifiable and organized distributions of vorticity. In addition, the mixing process involves all mechanisms typically found in vortex dynamics, such as stretching, break-up, concatenation, and self-induction of vortices. Experimental work suggests that large-scale, time-dependent structures, with periods much longer than the time of an impeller revolution, are involved in many of the fundamental hydrodynamic processes in stirred vessels. For example, local velocity data histograms may be bi-modal or tri-modal, even though they are being analyzed as having only one mode in most Laser-Doppler experiments. Digital particle image velocimetry experiments have shown that large-scale asymmetries with periods up to several minutes exist in stirred vessels equipped with axial flow impellers. These complex phenomena are not limited to single-phase systems. Many industrial vessels are operated with a multiphase flow. In such systems, the gas holdup distribution may be asymmetric and oscillating. In solids suspension processes, solids can be swept from one side of the vessel to the other in an oscillating pattern, even in dilute suspensions. The numerical modeling of these complicated mixing processes is a daunting task. Direct numerical simulation (DNS) provides the most exact approach in which the mechanism involved in turbulent mixing can be accurately represented. DNS requires resolving the smallest eddies, which makes the approach prohibitively expensive, even with the most powerful computers of the present day and foreseeable future as well. On the other hand, the popular approaches based on Reynolds-Averaged Navier-Stokes (RANS) equations amount to averaging out the large eddies that are primarily responsible for mixing. One is left to model the effects of large eddies by relying on empirical data and phenomenological reasoning and hypotheses, which are often questionable. The advantage of Large Eddy Simulation (LES) is that it explicitly resolves the large eddies, which are responsible for much of the mass, energy, and momentum transport, and only small eddies are modeled with a sub-grid model. In this lecture we will first briefly review the fundamentals of turbulent flows in stirred vessels, and how modeling these has evolved during the past decade. The focus will be on those aspects of turbulence that are relevant to mixing processes and the modeling thereof. We will continue with a discussion of the applicability of various turbulence models. For single-phase systems, we will then discuss the application of LES to the prediction of large-scale chaotic flow structures in stirred tanks. The focus of those studies is on systems with unsteady flows that are especially difficult to model with eddy-viscosity style models, namely those with strong swirl such as glass-lined mixing vessels (which usually have one baffle) and multiple impeller systems with strong interaction between the impeller flows. For multiphase systems, turbulence modeling is an even greater challenge. Interesting developments in this field include the use of LES models coupled with discrete particle simulations. More recently, full Reynolds stress models for use on unstructured finite volume meshes for Eulerian-Eulerian multiphase flow models have become available. Recent results with these models and expected future developments will be discussed.