Development of Anisotropic Filtered Two Fluid Model Closures
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Filtered modelling of dynamic gas-particle flows has been actively studied by various groups around the world for more than a decade. Even so, the great complexity of this field of study means that several important knowledge gaps still exist. This thesis represents a significant step forward by closing several of the most important knowledge gaps through the development and rigorous assessment of new closures via detailed a priori and a posteriori analyses. The resulting set of filtered closures clearly outperforms the current state of the art, resulting in several valuable conclusions and recommendations. The primary conclusion from the present work is related to the critical importance of accounting for anisotropy in the filtered closures for drag and solids mesoscale stresses. For the filtered drag force, it was found that conventional isotropic closures strongly underpredict the drag correction in the directions perpendicular to gravity. A new formulation based on the drift velocity concept was found to account for this anisotropic effect in an efficient and natural manner. For the solids mesoscale stresses, the present work confirmed that the conventional approach based on the Boussinesq approximation results in large errors. In fact, studies showed that coarse grid simulations completely neglecting the solids mesoscale stresses perform better than those relying on the Boussinesq-based approach. Based on this knowledge, a new closure formulation was devised, conveniently allowing the prediction of the anisotropic solids mesoscale stresses via a single expression. Findings from the present study also challenged other conventions in the field. Firstly, the use of the filtered slip velocity as a second marker in the filtered drag force closure was found to lead to poor model performance. Secondly, a filter size to grid size ratio of unity appears to be the fundamentally correct ratio instead of the commonly employed ratio of 2. And thirdly, the 2D models derived in this work outperformed a 3D model from the literature in a validation study, suggesting that domain size independence of resolved simulations is more important than performing simulations in 3D. For reactive flows, the present work showed that a relatively simple closure can accurately predict the filtered reaction rate. In addition, the closure for the mesoscale species dispersion rate used in the filtered species transport equation was shown to have only a minor effect on reactor performance predictions. However, coarse grid reactive simulations were sensitive to the accuracy of the hydrodynamic filtered closures employed. Good hydrodynamic modelling is therefore the most important prerequisite for accurate large scale reactor performance predictions. Despite the progress made in this thesis, some important knowledge gaps persist. Firstly, this study did not attempt to quantify the generality of the proposed closures to flow situations with different particle and fluid properties. Such studies are required before the newly proposed closures can be recommended for use in reactors with particle and fluid properties that are very different from the FCC-type system considered in the present work. Secondly, an important effect related to the ratio of the domain width to the length of macro-clusters resolved in coarse grid simulations was identified. This effect required the use of a larger filter size to grid size ratio in narrow domains and further studies are required to find a general solution to this challenge. However, informed application of the anisotropic closures proposed in this thesis to real fluidized bed reactor problems can already be recommended. Experience from such studies can further accelerate the development of closures for filtered models towards the goal of their ubiquitous deployment for design, optimization and scale-up of fluidized bed reactors in industry.