Direct numerical simulation of near-wall turbulent flow and ionic mass transport in electrochemical reactors using a hybrid finite element/spectral method
Introduction
Turbulent flow in combination with mass and heat transfer is encountered in many chemical and electrochemical processes. For example, forced flow is used to enhance mass transport during electrolytic deposition of metal coatings. Fluid flow influences the concentration of ions at the electrode and hence influences the current density. As a result, local variations in flow can result in non-uniform metal deposition. In case of metal alloy plating, local flow variations can also result in non-uniform alloy composition. Temperature variations at electrodes are also important since they can lead to electrical “burning” and bubble formation. Therefore, the study of fluid flow in electrochemical reactors is of practical importance and 3D numerical simulation has become an essential tool for this purpose.
Turbulent flow between rotating co-axial cylinders in a Taylor–Couette (TC) apparatus has been studied intensively for decades since the pioneering works of Couette [1], Mallock [2], [3] and Taylor [4]. TC flow exhibits a wide variety of flow regimes which makes it an ideal subject for analysis. Due to curvature of the cylinders, instabilities develop in TC systems. With increasing Reynolds number, these instabilities lead to annular vortices called Taylor vortices (TV) and finally result in fully turbulent flow. Many experimental and theoretical results on TC flow (TCF) were obtained by Swinney and coworkers during last two decades, see [5] and references herein. The most recent numerical simulations of TCF are found in [6], [7].
Apart from its fundamental importance, TC flow has a lot of practical applications such as dynamic filtration, viscosimetry and turbulent chemical or electrochemical reactors. However, to use TC flow for electrochemical reactors, it is important to understand mass transport in this geometry. Experimental studies of transport phenomena in electrochemical processes on rotating cylinders were performed by different authors [8], [9], [10], [11], [12], [13], but only a few studies are devoted to 3D numerical modeling. A comprehensive study was performed in [14], [15] where Large Eddy Simulation (LES) modeling was used to calculate the fluid flow in a reactor with a planar geometry. Nevertheless, comprehensive analysis of electrochemical systems using combined modeling of the Navier–Stokes equations (NSE) and heat and mass transport equations is still lacking. Some studies related to turbulent flow and passive scalars are available in the literature (see [16] and references herein).
In principle, three different approaches of theoretical description of turbulent flow are possible. The first approach is based on the Reynolds-Average Navier–Stokes (RANS) method [17] where the effect of turbulence is introduced into the model by filtering (averaging) of NSE in time. The second approach is based on direct numerical simulation (DNS) of NSE with spatial and time resolution fine enough to capture all scales present in the flow. Although this is the most accurate method, its drawback is a steep growth of the number of degrees of freedom as the Reynolds number increases. The third approach is the LES method where small-scale turbulence is modeled via a subgrid model [18]. Although some subgrid models are well established and confirmed by numerous experiments [19], selection of the case-specific submodel is always based on a preliminary analysis of experimental data. In case of electrochemical systems, correct statistics of turbulence and power spectral density of the velocity gradient, concentration and temperature fluctuations are especially important for adequate evaluation of the local current density. Therefore imposing an inappropriate subgrid model may lead to wrong statistics which are not physical.
So both DNS and LES provide the accurate solution of the flow field in a fully turbulent regime if meshing is fine enough in the computational domain, especially in the near-wall regions. However there is a shortcoming which becomes a serious obstacle if real electrochemical reactors are analyzed. The Schmidt number for the majority of practical applications is about . At such Schmidt numbers, the boundary layer relevant to the ionic mass transport near the electrode is much thinner than the hydrodynamic boundary layer. So two different scales are observed in the system: the larger scale defined by the fluid flow and the smaller scale controlled by the ionic diffusion. Therefore even if the hydrodynamic boundary layer is properly resolved, the diffusion boundary layer requires a much finer mesh. Consequently simulation of ionic transport in electrochemical systems is a challenging problem due to different time and spatial scales of all relevant processes.
The goal of this research was twofold. The first objective was the development of a numerical tool for the analysis of mass transfer in purely turbulent regimes. In this paper a hybrid finite element method in meridian planes combined with a Fourier expansion in the azimuthal direction for the Navier–Stokes equations coupled with ionic and energy transport equations is first described. Similar hybrid FE/spectral discretization have already been proposed for the Navier–Stokes equation and magnetohydrodynamics equations. See e.g. [20], [21], [22], [23] and references therein. Therefore, the first originality of this paper lies in the addition of passive scalar transport to a hybrid FE/spectral discretization.
The second contribution of this paper is the application of this hybrid method to DNS of turbulent flows and ion transport in an electrochemical cell with validation with respect to experiments. As a result, the numerical tool presented in this work can assist or even replace experimental studies where data on mass transfer are difficult to determine by classical electrochemical methods.
This paper is organized as follows. Section 2 contains the description of the experimental setup and the characteristic numbers for the experimental geometry. It also describes the measurements of near wall turbulence and local current density. Section 3 provides the governing equations of the model and explains the details of the numerical implementation. Coupling between momentum, mass transport and energy transport equations is also shown. Section 4 is devoted to the analysis of the results with mean velocity profiles, velocity fluctuations, wall shear stress, average limiting current (velocity gradient) near the electrode surface and its spectral analysis. Comparison with experimental data is also provided. Section 5 gives analysis of the concentration field via simulation with a passive scalar solver. A full 3D simulation for a low Reynolds and Schmidt number is first presented, followed by a preliminary axisymmetric simulation under experimental conditions with high Reynolds and Schmidt number. The conclusions are drawn in Section 6.
Section snippets
Experimental setup and characteristic number evaluation
Fig. 1 shows the TC reactor which was used in the experimental part of the research. The dimensions of the reactor are given in Table 1. The different flow structures which are present in the transition regime to fully developed turbulence can be described by three dimensionless parameters which were identified by Chandrasekhar [24]. The first parameter is the ratio of the inner to outer cylinder diameter denoted by . The second one corresponds to the Reynolds number based on the gap
Governing equations
Momentum and mass transport is modeled using a hybrid finite-element/spectral method developed from [6], [28], [29], [30], [31]. The new method assumes periodicity in one direction so the spectral approach is used for discretization of the governing equations in the azimuthal direction. In the transverse direction, discretization is performed by means of finite elements in meridional 2D planes, allowing complex axisymmetric geometries. No subgrid model was applied so the modeling was a pure DNS
Near wall turbulence: comparison of model and experiment
A series of DNS simulations at different rotation speeds between 100 and 1000 rpm ( and correspondingly) was performed in this work. The finite element mesh in cross planes was stretched along the walls. It contained mesh nodes at rpm and nodes at rpm correspondingly. Along the rotating electrode the number of nodes was 128. The number of Fourier modes in the azimuthal direction was 128 or 256 at the same sets of rotation speeds. Typical
Modeling of ionic transport
The model Eqs. (1), (2), (3) have been used for the simulation of fluid flow and mass transfer at different rotation speeds. As a first validation test, full 3D simulations of the coupled fluid flow and concentration field were performed at the limit of moderate Reynolds numbers and small Schmidt number. A hypothetical electrochemical solution with the diffusion coefficient of m2/s was simulated at a Reynolds number of and a Schmidt number of , Fig. 13. Since the Schmidt
Conclusions
A hybrid FE/spectral method for direct numerical simulation of near-wall turbulent flow and ionic mass transport in electrochemical reactors was presented. The method accounts for periodicity or quasi-periodicity of the reactor geometry. It uses spectral decomposition in the azimuthal direction and FEM expansion in meridian planes of the flow defined by the radial and axial direction. As a result, the fluid flow can efficiently be calculated with a high level of parallel efficiency which is
Acknowledgments
The authors express their gratitude to Wouter Van de Putte for performing mass transport measurements and to David Vanden Abeele and Yves Detandt for assistance with the SFELES code. The research was financed by the Institute for the Promotion of Innovation by Science and Technology in Flanders through the MuTEch project (IWT, contract No. SBO 040092). M.R. was supported by a fellowship from FRIA (Fonds pour la formation à la Recherche dans l’Industrie et dans l’Agriculture) for this work. M.K.
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