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Benchmark Simulation of Laminar Reactive Micromixing Using Lattice Boltzmann Methods
Abstract
Micromixers are chemical processing devices with complex flow patterns applied for both mixing and reaction of chemical species. In current research, laminar reacting multicomponent flows are considered. Despite the laminar streaming regime (e.g., Re = 186), there exist secondary flow microstructures. For this setup, accurate predictions of those structures are possible with a large-eddy simulation on a fine mesh resolving till the Batchelor microscales. Utilizing the open-source lattice Boltzmann method (LBM) framework, OpenLB, a benchmark simulation of the reacting micromixer, is re-established with new, more precise computation results. In this context, a Schmidt-number-based stabilization method for LBM-discretized reactive advection–diffusion equations by laminar secondary flow structures is used. A convergence study is performed, which is also a novelty. All computations have been performed on the high-performance computing cluster HoreKa using up to 160 NVIDIA A100 graphics processing units.
... Hettel et al. [6] demonstrate LBM for calculations of multiple free jets in different configurations. Bukreev et al. [7] perform a benchmark simulation of a reacting micromixer considering reacting multicomponent flows with LBM. Montessori et al. [8] develop a thread-safe LBM numerical approach on GPU-based architectures and deploy it to simulate axisymmetric turbulent jets of a fluid evolving in a quiescent, immiscible environment. ...
Lattice Boltzmann methods provide a robust and highly scalable numerical technique in modern computational fluid dynamics. Besides the discretization procedure, the relaxation principles form the basis of any lattice Boltzmann scheme and render the method a bottom-up approach, which obstructs its development for approximating broad classes of partial differential equations. This work introduces a novel coherent mathematical path to jointly approach the topics of constructability, stability, and limit consistency for lattice Boltzmann methods. A new constructive ansatz for lattice Boltzmann equations is introduced, which highlights the concept of relaxation in a top-down procedure starting at the targeted partial differential equation. Modular convergence proofs are used at each step to identify the key ingredients of relaxation frequencies, equilibria, and moment bases in the ansatz, which determine linear and nonlinear stability as well as consistency orders of relaxation and space-time discretization. For the latter, conventional techniques are employed and extended to determine the impact of the kinetic limit at the very foundation of lattice Boltzmann methods. To computationally analyze nonlinear stability, extensive numerical tests are enabled by combining the intrinsic parallelizability of lattice Boltzmann methods with the platform-agnostic and scalable open-source framework OpenLB. Through upscaling the number and quality of computations, large variations in the parameter spaces of classical benchmark problems are considered for the exploratory indication of methodological insights. Finally, the introduced mathematical and computational techniques are applied for the proposal and analysis of new lattice Boltzmann methods. Based on stabilized relaxation, limit consistent discretizations, and consistent temporal filters, novel numerical schemes are developed for approximating initial value problems and initial boundary value problems as well as coupled systems thereof. In particular, lattice Boltzmann methods are proposed and analyzed for temporal large eddy simulation, for simulating homogenized nonstationary fluid flow through porous media, for binary fluid flow simulations with higher order free energy models, and for the combination with Monte Carlo sampling to approximate statistical solutions of the incompressible Euler equations in three dimensions.
The OpenLB project provides a C++ package for the implementation of lattice Boltzmann methods (LBM) that is general enough to address a vast range of transport problems, e.g. in computational fluid dynamics. The source code is publicly available and constructed in a well readable, modular way. This enables for a fast implementation of both academic test problems and advanced engineering applications. It is also easily extensible to include new physical models.
Major new features include new performance-optimized and GPU-enabled multi-lattice coupling as well as a new subgrid-scale particle system. See https://www.openlb.net/news/openlb-release-1-6-available-for-download/ for the full release notes.
We present the first top-down ansatz for constructing lattice Boltzmann methods (LBM) in d dimensions. In particular, we construct a relaxation system (RS) for a given scalar, linear, d-dimensional advection–diffusion equation. Subsequently, the RS is linked to a d-dimensional discrete velocity Boltzmann model (DVBM) on the zeroth and first energy shell. Algebraic characterizations of the equilibrium, the moment space, and the collision operator are carried out. Further, a closed equation form of the RS expresses the added relaxation terms as prefactored higher order derivatives of the conserved quantity. Here, a generalized (2d+1)×(2d+1) RS is linked to a DdQ(2d+1) DVBM which, upon complete discretization, yields an LBM with second order accuracy in space and time. A rigorous convergence result for arbitrary scaling of the RS, the DVBM and conclusively also for the final LBM is proven. The top-down constructed LBM is numerically tested on multiple GPUs with smooth and non-smooth initial data in d=3 dimensions for several grid-normalized non-dimensional numbers.
We establish the notion of limit consistency as a modular part in proving the consistency of lattice Boltzmann equations (LBEs) with respect to a given partial differential equation (PDE) system. The incompressible Navier--Stokes equations (NSE) are used as paragon. Based upon the hydrodynamic limit of the Bhatnagar--Gross--Krook (BGK) Boltzmann equation towards the NSE, we provide a successive discretization by nesting conventional Taylor expansions and finite differences. We track the discretization state of the domain for the particle distribution functions and measure truncation errors at all levels within the derivation procedure. Via parametrizing equations and proving the limit consistency of the respective families of equations, we retain the path towards the targeted PDE at each step of discretization, i.e. for the discrete velocity BGK Boltzmann equations and the space-time discretized LBEs. As a direct result, we unfold the discretization technique of lattice Boltzmann methods as chaining finite differences and provide a generic top-down derivation of the numerical scheme which upholds the continuous limit.
As micromixers offer the cheap and simple mixing of fluids and suspensions, they have become a key device in microfluidics. Their mixing performance can be significantly increased by periodically varying the inlet pressure, which leads to a non-static flow and improved mixing process. In this work, a micromixer with a T-junction and a meandering channel is considered. A periodic pulse function for the inlet pressure is numerically optimized with regard to frequency, amplitude and shape. Thereunto, fluid flow and adsorptive concentration are simulated three-dimensionally with a lattice Boltzmann method (LBM) in OpenLB. Its implementation is then combined with forward automatic differentiation (AD), which allows for the generic application of fast gradient-based optimization schemes. The mixing quality is shown to be increased by 21.4% in comparison to the static, passive regime. Methodically, the results confirm the suitability of the combination of LBM and AD to solve process-scale optimization problems and the improved accuracy of AD over difference quotient approaches in this context.
Multiple-relaxation-time (MRT) lattice Boltzmann methods (LBM) based on orthogonal moments exhibit lattice Mach number dependent instabilities in diffusive scaling.
The present work renders an explicit formulation of stability sets for orthogonal moment MRT LBM.
The stability sets are defined via the spectral radius of linearized amplification matrices of the MRT collision operator with variable relaxation frequencies.
Numerical investigations are carried out for the three-dimensional Taylor–Green vortex benchmark at Reynolds number 1600.
Extensive brute force computations of specific relaxation frequency ranges for the full test case are opposed to the von Neumann stability set prediction.
Based on that, we prove numerically that a scan over the full wave space, including scaled mean flow variations, is required to draw conclusions on the overall stability of LBM in turbulent flow simulations.
Further, the von Neumann results show that a grid dependence is hardly possible to include in the notion of linear stability for LBM.
Lastly, via brute force stability investigations based on empirical data from a total number of 22696 simulations, the existence of a deterministic influence of the grid resolution is deduced.
We present the OpenLB package, a C++ library providing a flexible framework for lattice Boltzmann simulations. The code is publicly available and published under GNU GPLv2, which allows for adaption and implementation of additional models. The extensibility benefits from a modular code structure achieved e.g. by utilizing template meta-programming. The package covers various methodical approaches and is applicable to a wide range of transport problems (e.g. fluid, particulate and thermal flows). The built-in processing of the STL file format furthermore allows for the simple setup of simulations in complex geometries. The utilization of MPI as well as OpenMP parallelism enables the user to perform those simulations on large-scale computing clusters. It requires a minimal amount of dependencies and includes several benchmark cases and examples. The package presented here aims at providing an open access platform for both, applicants and developers, from academia as well as industry, which facilitates the extension of previous implementations and results to novel fields of application for lattice Boltzmann methods. OpenLB was tested and validated over several code reviews and publications. This paper summarizes the findings and gives a brief introduction to the underlying concepts as well as the design of the parallel data structure.
The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection–diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods
to approximate different types of partial differential equations.
An overview of the mixing performances of micro T-mixers operating with a single fluid is presented. The focus is on the relationship between flow features and mixing. Indeed, T-mixers are characterized by a variety of regimes for increasing Reynolds numbers; they are briefly described, in particular in terms of the three-dimensional vorticity field, which can explain the different mixing performances. The effects of changes in the aspect ratios of the channels are also reviewed. The role of instability and sensitivity analyses in highlighting the mechanisms of onset of the different regimes is then described. These analyses also suggest possible geometrical modifications to promote mixing. We focus on that consisting in the downward tilting of the inlet channels (arrow-mixers). Arrow-mixers are interesting because the onset of the engulfment regime is anticipated at lower Reynolds numbers. Hence, the mixing performances of arrow-mixers with varying Reynolds number are described.
We study a multiple relaxation time lattice Boltzmann model for natural convection with moment-based boundary conditions. The unknown primary variables of the algorithm at a boundary are found by imposing conditions directly upon hydrodynamic moments, which are then translated into conditions for the discrete velocity distribution functions. The method is formulated so that it is consistent with the second order implementation of the discrete velocity Boltzmann equations for fluid flow and temperature. Natural convection in square cavities is studied for Rayleigh numbers ranging from 10^3 to 10^8. An excellent agreement with benchmark data is observed and the flow fields are shown to converge with second order accuracy.
We show that discrete lattice effects must be considered in the introduction of a force into the lattice Boltzmann equation. A representation of the forcing term is then proposed. With the representation, the Navier-Stokes equation is derived from the lattice Boltzmann equation through the Chapman-Enskog expansion. Several other existing force treatments are also examined.
A promising approach to quantify reaction rate parameters is to formulate and solve inverse problems by minimizing the deviation between simulation and measurement. One major challenge may become the non-uniqueness of the recovered parameters due to the ill-posed problem formulation, which requires sophisticated approaches such as regularization. This study investigates the feasibility of using spatially distributed reference data, i.e., concentration distributions of reactive flows, which could be obtained by magnetic resonance imaging (MRI), instead of isolated points or integral values to recover reaction rate parameters. We propose a combined framework of computational fluid dynamics (CFD) and gradient-based optimization methods, which minimizes the difference between the simulated concentration distribution and a given data set by automatic iterative parameter adjustments. The forward problem is formulated as a coupled system of reaction-advection-diffusion equations (RADE), which is solved by the lattice Boltzmann method (LBM). Therefore, a system of non-linear partial differential equations (PDE) acts as optimization constraints, limiting the possible outcomes of the inverse problem. A benchmark test case using a CFD simulation as reference data confirms the validity of the presented method by successfully identifying up to three a priori set reaction parameters reversely. With it, initial relative errors could be reduced from around 150% to 10^{−3} % in 13 optimization steps corresponding to 37 simulations. Even after reducing the accessible reference data from 2D concentration distributions to 1D outflow concentration distribution or by adding noise signals onto the reference data with a signal-to-noise ratio (SNR) of 5, our framework successfully recovered the parameters with a relative error of ≈ 1%. Both, the chosen LBM and optimization algorithms are implemented in the open-source library OpenLB.
Lattice Boltzmann methods (LBM) are well suited to highly parallel computational fluid dynamics simulations due to their separability into a perfectly parallel collision step and a propagation step that only communicates within a local neighborhood. The implementation of the propagation step provides constraints for the maximum possible bandwidth-limited performance, memory layout and usage of vector instructions. This article revisits and extends the work on implicit propagation on directly addressed grids started by A-A and its shift-swap-streaming (SSS) formulation by reconsidering them as transformations of the underlying space filling curve. In this work, a new periodic shift (PS) pattern is proposed that imposes minimal restrictions on the implementation of collision operators and utilizes virtual memory mapping to provide consistent performance across a range of targets. Various implementation approaches as well as time dependency and performance anisotropy are discussed. Benchmark results for SSS and PS on SIMD CPUs including Intel Xeon Phi as well as Nvidia GPUs are provided. Finally, the application of PS as the propagation pattern of the open source LBM framework OpenLB is summarized.
Nitrate removal in wastewater systems is crucially important to maintaining the quality of potable water as well as protecting vulnerable ecosystems. In this study, a denitrification model was incorporated into Lattice-Boltzmann model (LBM) to investigate denitrification efficacy of wastewater treatment in a static mixer-reactor. The model was used for simulating the effects of six solid and porous obstacle configurations, different inlet velocities, and inlet methanol concentrations on denitrification within a static mixer-reactor with six different configurations. The results show that all the obstacles and porous blocks could improve the denitrification efficiency. However, the performances of these configurations could be affected differently by the inlet conditions. At a low velocity (u=0.5u0), the denitrification rates are almost twice as that at u=1.5u0. However, at a high inlet velocity, there were small differences of the denitrification efficiency among different configurations. Furthermore, the rectangular obstacle had the greatest effects on the denitrification rates at the low inlet velocity while its performance at a higher velocity was almost comparable to those of the porous blocks. It is also found that at a high inlet methanol concentration (C2-u = 2.0C2-u0), all the configurations increased the denitrification efficiency of up to 10%, compared to clear channel. However, the rectangular obstacle resulted in a decrease of about 3% in denitrification at a low inlet methanol concentration (C2-u = 0.5C2-u0) while all other configurations could improve the performance. Although at a low methanol concentration, the solid obstacles could dwindle the performance, the porous blocks did enhance denitrification rates in all the inlet methanol concentrations. This implies that the porous configurations have a wider range of operational conditions, compared to other obstacle configurations. A proper design of static mixer-reactors without any moving part is able to improve the denitrification efficiency in wastewater treatment. The present model is a useful tool to evaluate the performance of different reactor configurations under combinations of all significant parameters, such as inlet velocity and methanol concentrations.
Traditional Lattice–Boltzmann modelling of advection-diffusion flow is affected by numerical instability if the advective term becomes dominant over the diffusive (i.e., high-Péclet flow). To overcome the problem, two 3D one-way coupled models are proposed. In a traditional model, a Lattice–Boltzmann Navier–Stokes solver is coupled to a Lattice–Boltzmann advection-diffusion model. In a novel model, the Lattice–Boltzmann Navier–Stokes solver is coupled to an explicit finite-difference algorithm for advection-diffusion. The finite-difference algorithm also includes a novel approach to mitigate the numerical diffusivity connected with the upwind differentiation scheme.
The models are validated using two non-trivial benchmarks, which includes discontinuous initial conditions and the case Peg→∞ for the first time, where Peg is the grid Péclet number. The evaluation of Peg alongside Pe is discussed. Accuracy, stability and the order of convergence are assessed for a wide range of Péclet numbers. Recommendations are then given as to which model to select depending on the value Peg—in particular, it is shown that the coupled finite-difference/Lattice–Boltzmann provide stable solutions in the case Pe→∞, Peg→∞.
Detailed numerical analyses of temperature and air velocity distributions are relevant to assess thermal comfort in a wide range of applications. Until now mainly simulations based on Reynolds-averaged Navier–Stokes equations (RANS) are used, whereby fluctuations as well as anisotropy of the turbulence are represented with insufficient precision. This paper applies a thermal large eddy lattice Boltzmann method (LES-LBM) as an efficient and accurate transient modeling of turbulence. The benchmark case Manikin Heat Loss for Thermal Comfort Evaluation is studied and the model of Predicted Mean Vote (PMV) is applied for estimating thermal sensation. The results for the air velocity, the temperature field and the PMV show a satisfactory agreement with both, the experiment and the results from RANS simulations. The accuracy and the model quality of the simulation are increased further by considering the buoyancy and an inlet seeding. This suggests a successful evaluation of the present model, whereby additional transient flow field data are provided. The obtained transient flow field data, however, motivates future work to study thermal comfort in the present manner. The investigation of the influence of fluctuations on thermal comfort as well as the application to more complex problems seem promising.
Mixing at microscales is purely governed by the diffusion mass transport phenomenon, which is a time‐consuming process requiring a prolonged length of the microchannel to obtain desired results. The present study proposes a novel three‐dimensional helical micromixer (TDHM) with a rectangular cross‐section to achieve splendid mixing performance within a short distance contrary to the simple T‐micromixer (STM). A thorough numerical investigation of mixing performance and fluid flow patterns has been conducted using the continuity, species transport, and the Navier–Stokes equations with Newtonian and non‐Newtonian fluid at a wide range of Reynolds number (0.2–320) and mass flow rate (0.00005–0.091 kg/h), respectively. Blood is selected as the non‐Newtonian fluid, and its rheological characteristics are numerically captured by implementing the Carreau–Yasuda model, whereas water is used to study mixing with the Newtonian fluid. At Re = 2, the mixing index of TDHM is 40. 5% more than that of the STM with water as the working fluid, whereas for blood, it is 34.3%, and thus, it was concluded that the TDHM gives much better performance at much less axial distance than that of the STM at all values of the Reynolds number and flow rates considered in the study. Therefore, TDHM can be utilized for various biomedical, chemical, and biochemical applications.
The CFD simulation of fast reactions in laminar flows can be computationally challenging due to the lack of appropriate sub-grid micromixing models in this flow regime. In this work, simulations of micromixing via the implementation of the competitive-parallel Villermaux/Dushman reactions in a T-micromixer with square bends for Reynolds numbers in the range 60-300 are performed using both a conventional CFD approach and a novel lamellae-based model. In the first, both the hydrodynamics and the concentration fields of the reaction species are determined directly using a finite volume approach. In the second, the hydrodynamic field from the CFD calculations is coupled with a Lagrangian model that is used to perform the chemical reactions indirectly. Both sets of results are compared with previously published experimental data and show very good agreement. The lamellar model has the advantage of being much less computationally intensive than the conventional CFD approach, which requires extremely fine computational grids to resolve sharp concentration gradients. It is a promising solution to model fast chemical reactions in reactors with complex geometries in the laminar regime and for industrial applications.
Computational fluid dynamics (CFD) is very appealing to investigate mixing and reaction in microdevices, as it allows easily investigating different operating conditions as well as mixer geometries. This latter aspect is very important as the flow in microdevices is laminar so the mixing between reactants should be promoted by a clever mixer design, aimed at breaking the flow symmetries. Recently time periodic motions that improve mixing have been observed to take place in a T‐junction at low Reynolds numbers. In this case the numerical modelling should be based on direct numerical simulations (DNS), thus involving high computational resources. In this work, two different CFD approaches, i.e., finite volume and spectral element methods, are applied and compared for the analysis of the mixing process in the well known T‐shaped micromixer. Spectral elements methods are particularly suited for DNS; however, they have been scarcely applied to study micromixers, while plenty of works can be found with finite volume methods. The analysis is carried out using both ideal and non‐ideal liquid binary mixtures, the latter presenting a negative fluidity of mixing (i.e., the viscosity of the mixture is higher than that of the pure components). Moreover the numerical results are validated with simple flow visualization experiments. This article is protected by copyright. All rights reserved
Microreaction engineering enables new strategies in process intensification. A precise analysis of local mass transfer and hydrodynamics in micromixers for different flow regimes are strongly needed for a complete understanding the processes occurring. One of the simplest in the manufacture, but at the same time, quite effective T-shaped micromixer was used for numerical investigation and analyzing the mixing quality and flow regimes as well as the influence of different fluids properties on this parameters. It was numerically revealed that the viscosities and the densities, as well as the initial temperatures and the rheology of mixing fluids have significant effects on the flow regimes and the mixing efficiency of two fluids. In this study viscosities and densities ratios of mixing fluids ranged from 1 to 2; the coefficient n in power-law model of non-Newtonian fluids ranged from 0.3 to 1; the initial temperatures difference of two fluids was varied up to 40 °C. Mixture components concentration as well as pressure and velocity fields distribution in the micromixer was obtained. The dependence of fluids mixing efficiency and the pressure drop, as well as a map of flow regimes and mixing modes on the Reynolds number and properties of miscible fluids was numerically established.
This book is an introduction to the theory, practice, and implementation of the Lattice Boltzmann (LB) method, a powerful computational fluid dynamics method that is steadily gaining attention due to its simplicity, scalability, extensibility, and simple handling of complex geometries. The book contains chapters on the method's background, fundamental theory, advanced extensions, and implementation.
To aid beginners, the most essential paragraphs in each chapter are highlighted, and the introductory chapters on various LB topics are front-loaded with special "in a nutshell" sections that condense the chapter's most important practical results. Together, these sections can be used to quickly get up and running with the method. Exercises are integrated throughout the text, and frequently asked questions about the method are dealt with in a special section at the beginning. In the book itself and through its web page, readers can find example codes showing how the LB method can be implemented efficiently on a variety of hardware platforms, including multi-core processors, clusters, and graphics processing units. Students and scientists learning and using the LB method will appreciate the wealth of clearly presented and structured information in this volume.
A novel passive chaotic advection micromixer with periodic nozzle-diffuser-like (NDL) obstacles is proposed. Lattice Boltzmann simulations of two incoming flows were performed for Reynolds numbers in the range and two Schmidt numbers and 50. The numerical study focuses on the evaluation of the mixing efficiency of the new design and the analysis of mass transfer as a function of the Péclet number. The flow was identified as spatially periodic. The results are compared with those for a plain rectangular channel. The asymmetric design and placement of the obstacles lead to splitting and recombination of the flows, promoting transverse flow. Two principal characteristics of chaotic advection were observed at high Pe: helical, intertwined streamlines and hyperbolic points in transverse planes. The mixing efficiency was found to increase with channel length, whereas for the proposed channel the mixing efficiency was observed higher than that for the rectangular channel for .
A methodology was developed to select adequate, commercially available micromixers for mixing sensitive chemical reactions. The range of flow rates can be derived at which the selected micromixers have to be operated to ensure the required mixing intensity. This methodology enables the selection of adequate micromixers for the scale up of the chemical reactions to higher flow rates. Two chemical test reactions were used for an experimental approach to characterize the selected microreactors. Both reactions are based on the effect of micromixing on the product distribution of competitive reaction systems. Flow rates and pressure drop were determined at which the mixing times are short relative to the reaction times. In this case, influences of mixing on the selectivity of the reference reaction can be neglected. Since two reference reactions with different time scales for mixing and reaction were tested, it was possible to study the mixing performance of a variety of micromixers over a wide range of flow rates. The investigated micromixers differ in their dimensions, internal geometry, and mixing principle. In the present work, overview tables are provided as a tool to evaluate the commercially available micromixers for specific applications. Further, the influence of mixing principle and pressure drop is discussed.
In microfluidic components, the mixing of material between streams in
the flow is primarily by molecular diffusion. As diffusive mixing is
very slow compared to the convection of fluid down the microchannel, the
length of the microchannel required to achive well-mixed streams is
prohibitively long. We have recently begun a joint computational and
experimental effort to design and build microreactors with enhanced and
controlled mixing and surface delivery capabilities. In this paper, we
discuss results of numerical simulations of liquid flow in a candidate
configuration, based on the staggered herringbone mixer reported by
Stroock et al. We compare the mixing efficiency of the herringbone mixer
with that of a standard rectangular microchannel. The numerical model
solves the incompressible Navier-Stokes equations coupled with the
equation for conservation of species. The simulations demonstrate that
the major effect of the herringbone structures is to deflect the flow
around them and upwards, which results in the formation of convective
rolls that increase the interface area between fluid components in the
center of the microchannel, and therefore result in enhanced mixing.
This work presented a multi-component LBM approach to model the scalar mixing process with the chemical reactions in a liquid system. Two-dimensional simulation was applied to investigate the enhancement of the mixing performance at the laminar flow regime with very low Re numbers. The mixing degree measured with both IOS (intensity of segregation, characterized by the pure mixing without reaction) and XS (segregation index, characterized by the chemical reaction) was used to quantify the mixing performance. The investigation in both pure mixing process and the reacting flow system showed that the flow-focusing effect with the simple geometry could significantly enhance the mixing as well as the subsequent chemical reactions at very low Re numbers. We expected the theoretical approach described in our study to provide an effective method for micro-device design and optimization.
This paper introduces and exploits a hybrid numerical approach for fully resolved numerical simulations of reactive mixing in T-shaped microreactors and thereby enables a computational analysis of how chemical reactions interact with convective and diffusive transport. The approach exploits the fast redirection of the flow inside the mixing channel, resulting in a flow field with positive axial flow component everywhere after a short entry zone. This allows handling the axial flow direction as a pseudo-time variable, so that the evolution of the concentration profile can be computed consecutively on successive cross sections, following the main axial flow direction. With this approach the finest length scales, given by the Batchelor length scale, can be resolved for such a reactive mixing process inside a T-microreactor at stationary flow conditions. This allows for a detailed analysis of the mixing state as well as important characteristics of the reactive mixing process like yield and selectivity. The concrete numerical simulations yield local diffusion times inside the reactor, reveal the influence of the strength of the secondary flow on the progress of the chemical reaction and show how local selectivities result from the species transport.
The most widely used definitions of a vortex are not objective: they identify different structures as vortices in frames that rotate relative to each other. Yet a frame-independent vortex definition is essential for rotating flows and for flows with interacting vortices. Here we define a vortex as a set of fluid trajectories along which the strain acceleration tensor is indefinite over directions of zero strain. Physically, this objective criterion identifies vortices as material tubes in which material elements do not align with directions suggested by the strain eigenvectors. We show using examples how this vortex criterion outperforms earlier frame-dependent criteria. As a side result, we also obtain an objective criterion for hyperbolic Lagrangian structures.
Rapid processes such as certain organic reactions or precipitations at high supersaturation require the rapid mixing provided by jet mixers. Micromixing in a confined impinging jets (CIJ) mixer was characterized employing the Damköhler number to correlate processing. A scaling theory for the characteristic micromixing time, proportional to momentum diffusion starting at the Kolmogorov microscale, is shown as sufficient to express the micromixing performance of the CIJ mixer. A recently characterized second-order competitive reaction set is used as a “chemical ruler” to assign an absolute value to the mixing time in the CIJ mixer. A wide range of characteristic time (320 to 5 ms) is evaluated with hydrochloric acid competing for sodium hydroxide neutralization or 2,2-dimethoxypropane acid catalyzed hydrolysis. This reaction set was sensitive enough to detect the onset of a turbulent-like flow at a Re of 90 and was able to demonstrate a decrease in undesired products up to the highest Re tested, 3,800 or a jet velocity of 19 m/s. It represents a significant advancement to the reaction sets and techniques used for previous mixing studies, which are reviewed. Experiments verify the characteristic mixing time in a CIJ mixer scales as the inverse of the jet velocity to the three halves power, and the “mesomixing volume” (the volume over which the majority of flow energy was dissipated) is best approximated as proportional to the internozzle separation cubed. For each of the different jet diameters, chamber diameters and outlet configurations tested, the selectivity of the reaction scaled linearly with the Damköhler number, as determined from the known reaction kinetics and the calculated Kolmogorov diffusion time. The first full characterization is provided of micromixing in impinging jets that allows the prediction of mixing performance, reaction selectivity, and scale-up criteria.
When some external agency imposes on a fluid large-scale variations of some dynamically passive, conserved, scalar quantity θ like temperature or concentration of solute, turbulent motion of the fluid generates small-scale variations of θ. This paper describes a theoretical investigation of the form of the spectrum of θ at large wave-numbers, taking into account the two effects of convection with the fluid and molecular diffusion with diffusivity k. Hypotheses of the kind made by Kolmogoroff for the small-scale variations of velocity in a turbulent motion at high Reynolds number are assumed to apply also to small-scale variations of θ.
We extend and apply a method for the numerical computation of convective and diffusive mixing in liquid systems with very fast irreversible chemical reaction to the case of unequal diffusivities. This approach circumvents the solution of stiff differential equations and, hence, facilitates the direct numerical simulation of reactive flows with quasi-instantaneous reactions. The method is validated by means of a neutralization reaction which is studied in a T-shaped micromixer and compared with existing experimental LIF-data. Because of their large are-to-volume ratio, microreactors are well suited for fast chemical reactions which are seriously affected by the slow diffusive transport in aqueous media. Numerical computations for different reactor dimensions reveal the fact that, in a dimensionless setting, the obtained conversion is independent of the reactor size, if the flow conditions are the same. This corresponds to an increase of space-time-yield proportional to the square of the inverse scale factor. © 2009 American Institute of Chemical Engineers AIChE J, 2009
Making chemical processes safe requires a thorough knowledge of the kinetic and thermal parameters of the chemical reactions involved.The aim of this work was to develop a calorimetric method particularly adapted to the study of fast exothermal reactions. The proposed system combines a microreactor with a commercially available microcalorimeter.The microreactor was inserted into the cavity of the commercial calorimeter and the thermal efficiency of the system was optimized. The flow in the reaction channel of the microreactor was found to be purely laminar and the mixing time corresponded to the time for radial diffusion. Due to the small size of the channels, the mixing time was found to be adequate and not limiting for the characterization of fast reactions. First, a model reaction was studied in order to validate the results obtained with the microsystem and to avoid the risk of systematic errors. In a second stage, a previously unknown fast exothermal reaction was characterized. The heat flows measured during the reaction reached 160 000 W kg−1 but the conditions, however, remained completely isothermal. The global kinetics of this reaction as well as its activation energy were determined.
Successful applications of the chemical microprocess technology in the field of screening and the synthesis of fine chemicals using also new preparation routes are presented. By means of these examples it is shown that the microprocess technology allows new screening methods for the preparation of compound libraries or the investigation of catalysts, breaks new ground for the preparation of fine chemicals and offer novel possibilities for the production of chemicals on an industrial-scale.
In relation to the time-scale of chemical kinetics, diffusive transport in micro-devices is faster than in conventional mixers. To exploit the resulting potential for chemical process engineering, size effects evident in the transport processes have to be understood. For this purpose, the scaling behaviour concerning the transport of mass, momentum and heat are considered. Just as much, the mixing behaviour of flow mixers on micro-scales needs to be further investigated. Therefore, based on numerical simulations, the mixing characteristic of a T-shaped micro-reactor with rectangular cross sections is studied for three different flow regimes. For the description of the mixing quality, Danckwerts’ intensity of mixing is complemented by a measure of the scale of segregation which employs the concept of specific contact area. To assess the efficiency of mixing in ducts, the cross directional contribution of the energy dissipation rate is defined. These concepts are applied to a T-shaped micro-mixer, employing high-resolution CFD-simulations. Furthermore, comparison with given experimental data is performed and shows remarkable agreement.
A kinetic theory approach to collision processes in ionized and neutral gases is presented. This approach is adequate for the unified treatment of the dynamic properties of gases over a continuous range of pressures from the Knudsen limit to the high-pressure limit where the aerodynamic equations are valid. It is also possible to satisfy the correct microscopic boundary conditions. The method consists in altering the collision terms in the Boltzmann equation. The modified collision terms are constructed so that each collision conserves particle number, momentum, and energy; other characteristics such as persistence of velocities and angular dependence may be included. The present article illustrates the technique for a simple model involving the assumption of a collision time independent of velocity; this model is applied to the study of small amplitude oscillations of one-component ionized and neutral gases. The initial value problem for unbounded space is solved by performing a Fourier transformation on the space variables and a Laplace transformation on the time variable. For uncharged gases there results the correct adiabatic limiting law for sound-wave propagation at high pressures and, in addition, one obtains a theory of absorption and dispersion of sound for arbitrary pressures. For ionized gases the difference in the nature of the organization in the low-pressure plasma oscillations and in high-pressure sound-type oscillations is studied. Two important cases are distinguished. If the wavelengths of the oscillations are long compared to either the Debye length or the mean free path, a small change in frequency is obtained as the collision frequency varies from zero to infinity. The accompanying absorption is small; it reaches its maximum value when the collision frequency equals the plasma frequency. The second case refers to waves shorter than both the Debye length and the mean free path; these waves are characterized by a very heavy absorption.
Die mehrskalige Analyse des laminaren reaktiven Flüssig-flüssig-Strömungsmischens mittels numerischer Simulationen in T-förmigen Mikroreaktoren erfordert die vollständige Auflösung aller relevanten Längen- und Zeitskalen. Diese erstrecken sich über mehrere Größenordnungen, beginnend bei den charakteristischen Abmessungen des Mikroreaktors von unter einem Millimeter mit der damit einhergehenden hydrodynamischen Verweilzeit und endend bei den Atomabständen im Submikrometerbereich mit den dazu gehörenden Reaktionszeiten. Zur Auflösung dieser Längen- und Zeitskalen wird das reaktive Strömungsmischen mittels der Finite-Volumen-Methode über eine Kombination aus 3D- und 2D-Simulationen untersucht: In den Bereichen des T-Mikroreaktors, in denen die stationäre Strömung nicht eindeutig gerichtet ist und Rückströmungen auftreten, werden auf dreidimensionalen blockstrukturierten Gittern die Navier-Stokes-Gleichungen und die Speziesgleichungen gelöst. Sobald jedoch die Strömung eine eindeutige Fortschrittsrichtung aufweist, d. h. keine Rückströmungen mehr auftreten, wird eine parabolisierte Form der Speziesgleichung auf zweidimensionalen Gittern gelöst. Die zu Grunde liegende Gleichung wird aus der stationären Speziesbilanz unter Vernachlässigung der axialen Diffusion abgeleitet. Unter homogenen Neumann-Randbedingungen bzgl. der Querschnittsrichtungen wird dabei die axiale Hauptströmungsrichtung zu einer Art Zeitvariablen, entlang der die Evolution des Systems geschieht. Diese Kombination aus 3-D- und 2-D-Simulationen erlaubt es, die notwendigen Gitterauflösungen für den gesamten T-Mikroreaktor zu erreichen. Validiert wird dieses Vorgehen durch den Vergleich von aus Simulationen und Experimenten erhaltenen Konzentrationsfeldern des Chelats [Cafluo4], der im T-Mikroreaktor durch die Reaktion von örtlich getrennt aufgegebenen Ca(II)- und Fluo4-Ionen gebildet wird. Neben dieser quasi-irreversiblen Reaktion zweiter Ordnung werden weitere Reaktionssysteme schneller und instantaner Reaktionen untersucht. Insbesondere sehr schnelle, d. h. annähernd instantane Reaktionen, führen dabei zu extrem steifen Differentialgleichungssystemen. Die numerische Simulation solcher Reaktionssysteme gelingt jedoch mit einem aus Arbeiten von Toor abgeleiteten Ansatz, der darauf basiert, dass die Edukte bei einer unendlich schnellen irreversiblen Reaktion nicht koexistieren. Die Möglichkeit, schnelle und instantane chemische Reaktionen numerisch zu handhaben, erlaubt die gezielte Analyse des Einflusses der Strömungsgeschwindigkeit und der sich damit ausbildenden überlagerten Sekundärströmung auf das Umsatzverhalten. Das durch eine Erhöhung der Reynolds-Zahl verbesserte konvektive Mischen führt zu einer Verkleinerung der Längenskalen und damit zu einem gesteigerten Umsatz bei transportlimitierten chemischen Reaktionen. Allerdings zeigen numerische Simulationen unter Variation der Reynolds-Zahl bis an die Grenzen des stationären Strömungsbereichs, dass durch die sukzessive Verkürzung der hydrodynamischen Verweilzeit der Umsatz nicht beliebig gesteigert werden kann. Eine Umsatzsteigerung kann unter Beibehaltung der Reynolds-Zahl auch durch eine Verkleinerung der Reaktorabmessungen erreicht werden. Im Fall einer Reaktion zweiter Ordnung führt die Verkleinerungen der Abmessungen des Reaktors um einen Faktor b << 1 auf Grund des unterschiedlichen Skalierungsverhaltens der flächenbezogenen Transportprozesse und der volumenbezogenen chemischen Reaktion zu einer relativen Verlangsamung der Reaktion um den quadratischen Wert des Faktors b gegenüber den Transportprozessen. Dadurch führt unter gleichen hydrodynamischen Bedingungen eine Verkleinerung des Mikroreaktors zu deutlich gesteigerten Raumzeitausbeuten. In weiteren Simulationen, bei denen die Reaktionsgeschwindigkeitskonstante über mehrere Größenordnungen variiert wird, kann gezeigt werden, dass selbst bei der durch den Übergang auf den Mikroreaktor und die Steigerung der Strömungsgeschwindigkeit hervorgerufenen Beschleunigung des konvektiven Stofftransports eine Limitierung des Umsatzes durch die Transportprozesse beobachtet wird. Die beobachtete Abnahme der Selektivität eines Parallelreaktionssystems zweier Reaktionen zweiter Ordnung mit unterschiedlichen Reaktionszeiten bei zunehmender hydrodynamischer Verweilzeit lässt sich ebenfalls auf die inhomogene Vermischung in dem laminar durchströmten T-Mikroreaktor zurückführen. Insgesamt kann durch die erzielte Auflösung aller relevanten Zeit- und Längenskalen sowie durch Variation von Stoff- und Systemeigenschaften das wechselseitige Zusammenspiel von konvektivem und diffusivem Stofftransport und schnellen chemischen Reaktionen hinsichtlich Umsatz, Ausbeute und Selektivität detailliert untersucht werden. Die ermöglicht ein tiefer gehendes Verständnis des Zusammenspiels der bei schnellen Reaktionen ablaufenden Transport- und Reaktionsprozesse und darauf aufbauend letztlich die Optimierung des reaktiven Strömungsmischens. Due to the large area-to-volume ratio micro-reactors exhibit an effective mass and heat transfer. They also show immediately responses to changes in operating conditions. Therefore micro-reactors are suitable for the analysis of fast exothermic chemical reactions. The multiscale analysis of laminar mixed reaction systems requires the full resolution of all relevant time scales and length scales. In a T-shaped micro-reactor the largest length scales are given by the dimensions of the reactor itself whereby the smallest lengths scales are equal to the interatomic distances. The related time scales are the residence time and the reaction time. Using the finite volume method the full resolution of all length scales is obtained by a combination of simulations on a 3-D and a 2-D grid: the transport equations for mass, momentum and species concentration are solved on a 3D grid as long as the flow of the aqueous solvent shows no distinct direction but areas, where backflow occurs. Once no further backflow emerges, the reorientation of the flow towards the outlet of the straight micro channel allows the handling of the axial flow direction as a pseudo-time variable, so that the evolution of the concentration profile can be computed on successive 2-dimensional cross sections, thereby following the main axial flow direction. The conversion of the axial flow direction into a pseudo-time leads to the parabolised version of the species equation. This equation is derived from the stationary species equation under the neglect of the diffusion in the mean flow direction. The model is complemented by homogeneous Neumann boundary conditions concerning the species distribution. Investigating the species distribution of the fluorescent complex [CaFluo4] which is formed in a reaction of second order between the Ca(II) and the Fluo4 ions the described numerical approach is validated by experimental data. Beside the quasi-irreversible formation of [CaFluo4] further reaction systems of second order are investigated. Thereby numerical simulations of very fast i. e. quasi-instantaneous chemical reactions lead to the problem of stiff differential equation systems. To avoid such numerically unmanageable equation systems an approach described by Toor is developed further, whereby the impossibility of a local coexistence of both educts is exploited. The developed methods to handle fast and quasi-instantaneous reaction systems facilitate the detailed analysis of the influence of an increased flow rate on the yield and selectivity of reactions of second order. The numerical simulations reveal the enhancement of the yield and selectivity due to an increased but still stationary and laminar flow rate. However, the analysis of the conversion also shows that it is not possible to ameliorate continuously the conversion by the acceleration of the flow. Due to the accompanying reduction of the residence time a maximum conversion can not be exceeded without leaving the range of a stationary laminar flow. An increase of the conversion and the selectivity is also achievable by the miniaturisation of the micro-reactor itself under constant flow conditions. Downsizing the dimensions of the micro-reactor by a factor of b << 1 a reaction of second order is accelerated relatively to the transport processes due to the different scaling behaviour of transport processes and chemical reactions. An increase of the space-time-yield by a factor b multiplied by b is found. Varying the rate constant of a reaction of second order it is revealed that even the changeover to a micro-reactor and the acceleration of the flow rate lead to a limitation of the conversion. In case of a reaction system with side reactions the noticed residence time dependent decrease of selectivity can be lead back to an inhomogenous mixing which causes local excess of those species, which react in the undesirable side reactions. In summary, the full resolution of all relevant time scales and length scales as well as the variation of boundary conditions and species characteristics allows a detailed analysis of the interactions of transport processes and fast chemical reactions. The influences on different yield and selectivity of a reaction system can be revealed. This enables an explicit comprehension of the reactive mixing and opens the access to further optimizations.
- Wörz O.
Computergestützte Analyse des reaktiven Strömungsmischens in T-Mikroreaktoren
- Lojewskia
Theoretische und Numerische Untersuchung des Strömungsmischens in Einem T-Förmigen Mikromischer
- Stemichc
Lattice Boltzmann Methods for Partial Differential Equations,” Doctoral Dissertation
- Simoniss
On the Application of the Eddy Viscosity Concept in the Inertial Sub-Range of Turbulence
- D K Lilly
Lilly, D. K., "On the Application of the Eddy Viscosity Concept in the
Inertial Sub-Range of Turbulence," NCAR Manuscript, Vol. 123,
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https://doi.org/10.5065/D67H1GGQ