Assessment of k–ε models using tetrahedral grids to describe the turbulent flow field of a PBT impeller and validation through the PIV technique

Abstract

In turbulence modeling, the RNG and Realizable models have important improvements in the turbulent production and dissipation terms in comparison to the Standard. The selection of the appropriate turbulence model has an impact on the convergence and solution in STRs, and they are used in mixing, multiphase modeling or as starting solution of transient models as DES and LES. Although there are several studies with the pitched blade turbine (PBT) impeller, most of them used the Standard model as representative of all k–ε models, using structured hexahedral grids composed of low number of cells, and in some cases under axial symmetry assumptions. Accordingly, in this work the assessment of the Standard, RNG and Realizable models to describe the turbulent flow field of this impeller, using the Multiple Reference Frame (MRF) and Sliding Mesh (SM) approaches with tetrahedral domains in dense grids, is presented . This kind of cell elements is especially suitable to reproduce complex geometries. Flow velocities and turbulent parameters were verified experimentally by PIV and torque measurements. The three models were capable of predicting fairly the pumping number, the power number based on torque, and velocities. Although the RNG improved the predictions of the turbulent kinetic energy and dissipation rate, the Realizable model presented better performance for both approaches. All models failed in the prediction of the total dissipation rate, and a dependence of its value on the number of cells for the MRF was found.

Introduction

Mixing in stirred tank reactors (STR) is one of the most important topics of chemical process engineering. Good mixing assures higher reaction efficiency and shorter process time, reducing process costs [1], [2]. Mixing in a STR relies on several geometrical parameters, configurations and operational conditions, but undoubtedly, impeller performance is the most influential factor of them all [3], [4], [5]. Therefore, careful analysis of the fluid flow driven by the impeller along the whole vessel is necessary to evaluate reactor performance. A large variety of impeller geometries have been developed to solve all sorts of industrial problems [6], [7], [8]. For its enhancement capabilities blending/mixing and heat transfer, one of the most studied impellers is the pitched blade turbine (PBT). Since the micro-mixing and macro-mixing rely on the flow velocities and turbulent components, suitable analytical, experimental and numerical tools are required to depict the overall flow patterns. In this sense, the solutions obtained using computational fluid dynamics (CFD) require a turbulence model capable of reproducing fairly the fluid behavior with a minimum computational effort. Reynolds-Averaged Navier–Stokes (RANS) models of the k–ε family represent a practical balance of the aforementioned constraints, since most powerful techniques as LES and DNS need an important amount of grid elements and simulation time, and thus, they are not suitable for practical engineering applications [9]. Several works have been devoted to analyze the flow inside a STR using different grid resolutions, spatial discretization, coupling scheme and turbulence model, either for PBT or Rushton turbines, using the Multiple Reference Frame (MRF) approach [10], [11], [12] or Sliding Mesh (SM) [6], [13], [14]. In such works, it is stated that the numerical models deliver fair predictions for mean velocities, but under-predicted values for the turbulent quantities (turbulent kinetic energy “k”, and dissipation rate “ε”), compared with the LDV or 2D PIV measurements. For example, Jaworski and Zakrzewska [15], performed a comparison of the Standard, RNG, Realizable and other models in a STR for a non-structured grid composed of 112,000 hexahedral elements, and subpredicted values were found for k. Murthy and Joshi [16], evaluated different turbulence models and impeller designs in domains composed of about 575,000 elements. According to their results, the Standard model performed weakly in the prediction of the turbulent production and dissipation profiles, specially near the blade regions where the flow is anisotropic. However, it is not clear if the zero thickness wall assumption adopted by the authors affected their predictions, as it is known that blade thickness influences the flow parameters [17]. Further, the Standard model was taken as the representative case for the k–ε family. Joshi et al. [18], [19], analyzed velocities and turbulent profiles induced by Rushton and PBT impellers using different turbulence models. The authors reported important deviations between the RNG and Standard model predictions for velocities, turbulent kinetic energy production and dissipation rates, and suggested that the Standard model performs well in unidirectional flow regions with weak recirculation. From the numerical point of view, at that time, the computational resources were limited in comparison to actual capabilities, and the grids solved were commonly of about half million cells or less, and were composed of hexahedral elements [11], [14], [20], [21]. Since grid refinement and cell number are important to capture the main flow gradients, especially for the turbulent parameters [22], it is desirable to assess those quantities using larger grids. Also, because the actual STRs contain accessories or their geometry is complex enough to be mapped by hexahedral grids, tetrahedral cells become an alternative, and its assessment becomes an interesting issue to address. Although this kind of cells has been proven with success in the CFD modeling of a STR agitated by impellers in complex geometries [23], [24], [25], there is no information about the performance of the different models under the same comparative basis. It must be taken into account that the different closure models Standard, RNG or Realizable obey to deep physic assumptions and care must be taken regarding generalizing the results of the Standard model as unique and representative for all closure scenarios. The importance of achieving good quality results from RANS simulations relies not only on the necessity of fair average quality predictions of mean flow velocity and turbulent parameters, but also, in the necessity of good starting solutions for unsteady approaches, to decrease the time required to reach the desired statistic convergence. Furthermore, the engineering modeling of multiphase flows, combustion, and thermally affected boundary layers still needs an adequate choice of a k–ε model to account for turbulence, as more sophisticated models (DNS, LES) demand a significant amount of computational resources. Based on the above discussion, the objective of this work is to assess the prediction capabilities of the Standard [26], RNG [27] and Realizable [28] k–ε models of the flow features in a STR with four baffles, driven by a 4 blade PBT impeller at Re ≈ 52,000, using unstructured grids. The validation is accomplished via estimation of the dissipation rate from torque measurements, and by 2D-PIV measurements of the flow field velocities and turbulent quantities near the impeller, when the full blade coincides with the measurement plane at 0°. This region was selected as it is the zone where the mean and turbulent energy are produced and transported, defining the subsequent flow patterns. Also, the performance of the Sliding Mesh (SM) method was assessed for the three turbulence models. The importance of this work, relies on the understanding of the models behavior in non-structured tetrahedral grids especially useful to represent complex geometries, which are the natural evolution of second generation STR devices.