Maxwell-Stefan Diffusion Equations Research Articles

A multiscale thermodynamic generalization of Maxwell-Stefan diffusion equations and of the dusty gas model

Despite the fact that the theory of mixtures has been part of non-equilibrium thermodynamics and engineering for a long time, it is far from complete. While it is well formulated and tested in the case of mechanical equilibrium (where only diffusion-like processes take place), the question how to properly describe homogeneous mixtures that flow with multiple independent velocities that still possess some inertia (before mechanical equilibrium is reached) is still open. Moreover, the mixtures can have several temperatures before the temperatures relax to a common value. In this paper, we derive a theory of mixtures from Hamiltonian mechanics in interaction with electromagnetic fields. The resulting evolution equations are then reduced to the case with only one momentum (classical irreversible thermodynamics), providing a generalization of the Maxwell-Stefan diffusion equations. Then, we reduce that description to the mechanical equilibrium (no momentum) and derive a non-isothermal variant of the dusty gas model. These reduced equations are solved numerically, and we illustrate the results on efficiency analysis, showing where in a concentration cell efficiency is lost. Finally, the theory of mixtures identifies the temperature difference between constituents as a possible new source of the Soret coefficient. For the sake of clarity, we restrict the presentation to the case of binary mixtures; the generalization is straightforward.

NUMERICAL RESEARCH OF THE CHANGE OF REGIME FOR UNSTABLE MASS TRANSFER IN A TERNARY GAS MIXTURE HYDROGEN- NITRIC OXIDE-NITROGEN

On the basis of the software package "MathCad", by solving the Stefan-Maxwell diffusion equations, the evolution of the features of mass transfer in a three-component gas mixture, depending on pressure changes, has been numerically studied. In this analysis, the mixing process is studied in a vertical cylindrical channel of a finite size and at the isothermal conditions. The governing equations are solved at the boundary conditions assuming the absence of matter transfer through the walls of diffusion channel. Through the Rayleigh partial numbers, the influence of the pressure change on the behaviour of diffusion and convective flows is examined. The numerical results reveal that an increase in the pressure leads to a change of modes in ternary gas mixture. The present results are in good agreement with the existing experimental data.

Special modes of diffusion mass transfer in isothermal triple gas mixtures

In isothermal three-component gas mixtures in quasi-stationary conditions the procedure of numerical solution of Stefan-Maxwell diffusion equations using the software package “MathCad” is considered. It is shown that one of the specific features of multicomponent diffusion is the appearance of nonlinear concentration distributions of components in vertical channels. Due to the difference in the coefficients of mutual diffusion of gases in some systems under certain conditions, nonlinear distributions of component concentrations lead to a non-monotonic distribution of the density of the gas mixture. By the example of hydrogen – nitrogen – methane and methane – n-butane – nitrogen systems at a constant temperature T = 298 K, the influence of the content of the component with the highest molecular weight in the triple gas mixture on the degree of nonlinearity of the density distribution in the diffusion channel is analyzed. It is suggested that the inversion of the density distribution of the mixture may be the cause of convective instability in triple gas systems. The results of calculations are compared with experimental data.

Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

This article proposes adaptive iterative splitting methods to solve Multiphysics problems, which are related to convection–diffusion–reaction equations. The splitting techniques are based on iterative splitting approaches with adaptive ideas. Based on shifting the time-steps with additional adaptive time-ranges, we could embedded the adaptive techniques into the splitting approach. The numerical analysis of the adapted iterative splitting schemes is considered and we develop the underlying error estimates for the application of the adaptive schemes. The performance of the method with respect to the accuracy and the acceleration is evaluated in different numerical experiments. We test the benefits of the adaptive splitting approach on highly nonlinear Burgers’ and Maxwell–Stefan diffusion equations.

Modeling of Plasma Dispersion Process in Vacuum Interrupters During Postarc Interval Based on FEM

This paper presents a novel finite-element modeling of interelectrode plasma dispersion process inside a vacuum interrupter (VI) prototype. A new set of coupled equations considered for the plasma simulation process consisting of the Stefan-Maxwell diffusion equations, collision and interaction between particles considerations, and Coulomb integral for conductivity calculation of plasma are utilized to give a better understanding from complicated vacuum plasma dispersion process inside a VI. Using SolidWorks software, an exact 3-D schematic of the prototype is drawn and imported to the proper finite-element method software. Considering suitable boundary conditions and proposed plasma equations set, including 10 kV/μs. The ramp function as a transient recovery voltage inserted to the contacts, the model is dynamically simulated, and the derived results have been discussed in detail. In continue, postarc current waveform is derived using the Maxwell equations. This result is compared to a practical-based common model which is named the Andrews-Varey transition model. The derived postarc waveforms are compared in a table. The similarities of results, for example, postarc time (difference between two models is less than 8%) or maximum value of postarc current (difference <;7.5%), ensure the accuracy of the new proposed plasma modeling strategy.

A robust lattice Boltzmann method for parallel simulations of multicomponent flows in complex geometries

Abstract In this work, we are concerned with robust parallel simulations of multicomponent flows in complex geometries by an extended lattice Boltzmann method. In Zudrop et al. (2014) [22] we presented a model for the incompressible Navier–Stokes and Maxwell–Stefan diffusion equations. Here, we prove that the implicit-to-explicit variable transformation of the model is well-posed under weak constraints on the dimensionless Maxwell–Stefan diffusivities. Furthermore, we analyze various boundary conditions for the multicomponent lattice Boltzmann model. These results make the model robust and well suited for complex geometries, which allows us to study realistic, large-scale mass transport applications.

A Model for Analysis of the Porous Nickel Electrode Polarization in the Molten Carbonate Electrolysis Cell

It is important to know the electrode kinetics of hydrogen production as well as to understand the effect of the mass transport in the gas phase for the analysis of the molten carbonate electrolysis cell (MCEC). A one-dimensional model based on the Maxwell-Stefan diffusion equations was applied to predict the mass transfer behavior in gas phase of the porous nickel electrode in the MCEC, combined with equations describing the current distribution in the electrolyte phase. The model gave a fair match to the experimental polarization data of the Ni electrode for varied inlet gas compositions of H2O, CO2 and H2 between 10 and 40%. The model was also deployed to evaluate the effect of the water-gas shift reaction (WGSR). The fitted kinetic coefficients and electrode porosity differed in the case when including the WGSR compared to when not including the WGSR. In both cases the model was able to well describe the porous nickel electrode behavior in the molten carbonate electrolysis cell.

Iterative solvers for the Maxwell–Stefan diffusion equations: Methods and applications in plasma and particle transport

In this paper, we are motivated to discuss a model based on a local thermodynamic equilibrium, weakly ionized plasma-mixture model used for medical and technical applications in etching processes. For studying the model, we consider a simplified model based on the Maxwell–Stefan model, which describes multicomponent diffusive fluxes in the gas mixture. The MS model is more adequate to describe complex mixtures without dominating background species. Based on additional conditions to the fluxes, we obtain an irreducible and quasi-positive diffusion matrix. Such problems result in nonlinear diffusion equations, which are more delicate to solve as simpler standard diffusion equations with Fickian’s approach. Here, we propose an efficient explicit time-discretization method, which is embedded to a fast iterative solver for the nonlinearities. Such a combination of coupling discretization and solver methods allows to simulate the delicate nonlinear differential equations more effectively. We present the efficie...

The Maxwell-Stefan Diffusion Limit for a Kinetic Model of Mixtures

We consider the non-reactive elastic Boltzmann equation for multicomponent gaseous mixtures. We deduce, under the standard diffusive scaling, that well prepared initial conditions lead to solutions satisfying the Maxwell-Stefan diffusion equations in the vanishing Mach and Knudsen numbers limit.

Multi-component gas transport in micro-porous domains: Multidimensional simulation at the macroscale

For decades, models based on the Maxwell–Stefan diffusion equations have been used to predict mass transfer of multi-component gases in porous media. However, the validity of these models is restricted to transport processes that are dominated by diffusion and wall-friction. Therefore, a new computational model is presented, which is based on a comprehensive theory of multi-component gas transport. It comprises conservation equations of mass and momentum for each species contained in the gas mixture together with an energy balance for the mixture. The model is intended to be used for investigations on the macroscopic scale. Thus, the local volume averaging technique is applied and friction coefficients, which encompass the full range of Knudsen numbers, are used to describe the drag forces exerted by the pore structure. In the volume averaged energy equation the gas and solid phases are considered a pseudo-homogeneous medium. The resulting set of highly coupled transport equations is implemented in a commercial computational fluid dynamics program. For validation purposes, the mass transfer model is first used to simulate the isothermal diffusion problem in a Loschmidt tube. Then, the simultaneous mass and heat transfer in a porous duct with cooled walls is investigated. In both cases, the numerical results are in very good agreement with experimental data.

Molecular transport in nanopores: a theoretical perspective

Molecular transport in nanopores plays a central role in many emerging nanotechnologies for gas separation and storage, as well as in nanofluidics. Theories of the transport provide an understanding of the mechanisms that influence the transport and their interplay, and can lead to tractable models that can be used to advance these nanotechnologies through process analysis and optimisation. We review some of the most influential theories of fluid transport in small pores and confined spaces. Starting from the century old Knudsen formulation, the dusty gas model and several other related approaches that share a common point of departure in the Maxwell-Stefan diffusion equations are discussed. In particular, the conceptual basis of the models and the validity of the assumptions and simplifications necessary to obtain their final results are analysed. It is shown that the effect of adsorption is frequently either neglected, or treated on an ad hoc basis, such as through the division of the pore flux into gas-phase and surface diffusion contributions. Furthermore, while it is commonplace to assume that cross-sectional pressure is uniform, it is demonstrated that this violates the Gibbs-Duhem relation and that it is the chemical potential that essentially remains constant in the cross-section, as near-equilibrium density profiles are preserved even during transport. The Dusty Gas model and Maxwell-Stefan model for surface diffusion are analysed, and their strengths and weaknesses discussed, illustrating the use of conflicting choices of frames of reference in the former case, and the importance of assigning appropriate values for the binary diffusivity in the latter case. The oscillator model, developed in this laboratory, which is exact in the low density limit under diffuse reflection conditions, is shown to represent an advance on the classical Knudsen formula, although the latter frequently appears as a fundamental part of many transport models. The distributed friction model, also developed in this laboratory for the study of multi-component transport at any Knudsen number is discussed and compared with previous approaches. Finally, the outlook for theory and future research needs are discussed.

Modeling of Mass Transfer in Nonideal Multicomponent Mixture with Maxwell-Stefan Approach

The Intalox metal tower packing was used to simulate an industrial relevant extractive distillation column for purifying azeotropic multicomponent mixture. In order to explain the inconsistencies in the modeling of transfer process in nonideal multicomponent distillation column, a method was developed with equilibrium stage models (EQ) and non-equilibrium model (NEQ) incorporated with Maxwell-Stefan diffusion equations in the framework of AspenONE (r) simulator. Dortmund Modified UNIFAC (UNIFAC-DMD) thermodynamic model was employed to estimate activity coefficients. In addition, to understand the reason for the diffusion against driving force and the different results by EQ and NEQ models, explicit investigations were made on diffusion coefficients, component Murphree efficiency and mass transfer coefficients. The results provide valuable information for basic design and applications associated with extractive distillation.

Model based operation of emulsion polymerization reactors with evaporative cooling: Application to vinyl acetate homopolymerization

This work aims at maximizing the productivity of emulsion homopolymerization processes. A dynamic model of emulsion polymerization processes is extended by the inclusion of vaporization from the liquid phases in the reactor to the gaseous phase. The multi-component gas–liquid mass transfer phenomenon is described by the Maxwell–Stefan diffusion equations, which are solved by a special algorithm. A novel operation strategy is developed for running a reactor optimally with respect to batch time. This strategy is applied first to an industrial scale reactor, which is run without using evaporative cooling. Then, based on the extended model, controlled vaporization is included by which additional heat is removed from the reaction system. This makes it possible to extend the restrictions imposed by the limited heat removal of the cooling jacket considerably. Simulation results are presented for the homopolymerization of vinyl acetate in an industrial scale reactor operated in semi-batch mode. The results show that a significant amount of heat can be removed by evaporative cooling thus leading to higher productivity.

Influence of unequal component efficiencies on trajectories during distillation of a homogeneous azeotropic mixture

The overall objective of this work is to examine the influence of interphase mass transfer on the composition trajectories in homogeneous azeotropic distillation. A total of 38 experiments were carried out in a bubble cap distillation column operated at total reflux with the system: water–ethanol– tert-butanol. The experiments were carried out in the two regions on either side of the distillation boundary connecting the ethanol–water and t-butanol–water azeotropes. In order to model the composition trajectories, a rigorous nonequilibrium (NEQ) stage model is developed. The NEQ model incorporates the Maxwell–Stefan diffusion equations to describe the intraphase transfers in the vapor and liquid phases. The only adjustable parameter in the NEQ model is the size of the vapor bubbles on trays. A choice of a bubble diameter of 4 mm in the developed NEQ model gave the best agreement with the experimental results for all of the 38 experimental runs. The Murphree efficiencies of the constituents in the ternary mixture were found to be significantly different from one another for all the runs. In order to ascertain the influence of unequal component efficiencies on the column composition trajectories, the experimental results were also simulated with an equilibrium (EQ) stage model assuming a uniform, constant efficiency for all components on all the trays. The value of this constant efficiency for any experimental run was obtained by averaging the individual component efficiencies for all the three components on all the trays, calculated by the rigorous NEQ model. The predictions of the EQ model leads to significantly worse predictions of the column composition trajectories for each of the runs, when compared to the NEQ model. It is found that the column composition trajectories are significantly altered due to differences in the component efficiencies. From a design view point, it is shown that for the water–ethanol– tert-butanol system, the attainment of a desired ethanol purity in the top product may require significantly larger number of stages than that anticipated by the EQ model incorporating constant component efficiencies.

Composition Trajectories for Heterogeneous Azeotropic Distillation in a Bubble-Cap Tray Column: Influence of Mass Transfer

The overall objective of this work is to examine the influence of interphase mass transfer on the composition trajectories in heterogeneous azeotropic distillation. Experiments were carried out in a bubble cap distillation column operated at total reflux with the systems: water–ethanol–cyclohexane and water–acetone–toluene. The experiments were carried out in regions of the composition space such that liquid–liquid phase splitting occurred on some of the trays. In order to model the composition trajectories, a rigorous non–equilibrium (NEQ) stage model is developed. The NEQ model incorporates the Maxwell–Stefan diffusion equations to describe the various intraphase transfers. The developed NEQ model is in good agreement with the experimental results for both experimental systems. In sharp contrast, an equilibrium (EQ) stage model fails even at the qualitative level to model the experiments. For example, for the water–ethanol–cyclohexane system the EQ model anticipates distillation boundary crossing when this does not take place in practice. For the water–acetone–toluene system the EQ model does not anticipate distillation boundary crossing when this phenomena is found in the experiments. It is concluded that for reliable design of azeotropic distillation columns we must take interphase mass transfer effects into account in a rigorous manner.

Experimental Verification of the Maxwell-Stefan Formulation in Describing Composition Trajectories During Azeotropic Distillation

Experiments were carried out in a bubble cap distillation column operated at total reflux with the quaternary mixture: water (l)–ethanol (2)–methanol (3)–acetone (4). This system has a binary minimum-boiling azeotrope for the water–ethanol mixture and the distillation boundary is represented by a surface with its corners at pure acetone, pure methanol and the water–ethanol azeotrope. For certain starting compositions the measured distillation composition trajectories clearly demonstrate that crossing of the distillation boundary is possible. In order to rationalize the experimental results, the authors develop a rigorous nonequilibrium (NEQ) stage model, incorporating the Maxwell–Stefan diffusion equations to describe transfer in either fluid phase. The developed NEQ model anticipates the boundary crossing effects, and is in excellent agreement with a series of experiments. In sharp contrast, an equilibrium (EQ) stage model fails even at the qualitative level to model the experiments. The differences in the NEQ and EQ trajectories emanates from differences in the component Murphree efficiencies, which in turn can be traced to differences in the binary pair vapour phase diffusivities. It is concluded that for reliable design of azeotropic distillation columns interphase mass transfer effects must be taken into account in a rigorous manner.

Influence of interphase mass transfer on the composition trajectories and crossing of boundaries in ternary azeotropic distillation

Abstract This paper examines the influence of mass transfer on the composition trajectories in multicomponent azeotropic distillation. Simulations were carried out for three different ternary systems: methanol–isopropanol–water, water–ethanol–acetone, and water–methanol–methylacetate. Two different models were used to calculate the composition trajectories in a tray column: an equilibrium (EQ) stage model and a rigorous nonequilibrium (NEQ) stage model based on the Maxwell–Stefan diffusion equations. The simulations show that the EQ and NEQ model trajectories could follow different composition paths and could end up in completely different corners of the composition space. Furthermore, in all the three case studies the NEQ model trajectory was found to cross the distillation boundary even when the boundary is a straight line. The study has implications for the development of improved separation strategies.

The need for using rigorous rate-based models for simulations of ternary azeotropic distillation

Experiments were carried out in a bubble cap distillation column operated at total reflux with the system: water–ethanol–methylacetate. This system has two binary azeotropes (water–ethanol and water–methylacetate), which gives a simple distillation boundary connecting the two azeotropes. All experiments were restricted to the homogenous region without liquid phase splitting. For certain starting compositions the measured distillation composition trajectories clearly demonstrate that crossing of the distillation boundary is possible. In order to rationalize our experimental results, we develop a rigorous nonequilibrium (NEQ) stage model, incorporating the Maxwell–Stefan diffusion equations to describe transfer in either fluid phase and a fundamental description of tray hydrodynamics. The developed NEQ model anticipates the boundary crossing effects, and is in excellent agreement with a series of experiments carried out in different composition regions. In sharp contrast, an equilibrium (EQ) stage model fails even at the qualitative level to model the experiments. The differences in the NEQ and EQ trajectories emanates from differences in the component Murphree efficiencies, which in turn can be traced to differences in the binary pair vapor phase diffusivities Ðy,ij. It is concluded that for reliable design of azeotropic distillation columns we must take interphase mass transfer effects into account in a rigorous manner.

Crossing of boundaries in ternary azeotropic distillation: influence of interphase mass transfer

This paper examines the influence of mass transfer on the composition trajectories in multicomponent azeotropic distillation. Simulations were carried out for five different ternary systems: methanol — isopropanol — water, water — ethanol — acetone, acetone — chloroform — methanol, acetone — chloroform — ethanol, and water — cyclohexane — ethanol. Two different models were used to calculate the composition trajectories in a tray column: an equilibrium (EQ) stage model and a rigorous nonequilibrium (NEQ) stage model based on the Maxwell-Stefan diffusion equations. The simulations show that the EQ and NEQ model trajectories could follow different composition paths and could end up in completely different corners of the composition triangular space. Futhermore, in some of the case studies the NEQ model trajectory was found to cross the distillation boundary even with the feed on the convex side or when the boundary is a straight line. The study has implications for the development of improved separation possibilities.

Diffusion of binary mixtures across zeolite membranes: entropy effects on permeation selectivity

Diffusion of binary mixtures across zeolite membranes is strongly influenced by the sorption behaviour of the individual components. When the components in the mixture have significantly different molecular sizes, size entropy effects tend to favour the sorption of the smaller species at high molecular loadings. This is the case, for example, for adsorption of methane and n-butane in silicalite. For mixtures of linear and branched alkanes having the same number of carbon atoms, configurational entropy effects tend to favour the sorption of the linear alkane becuase these molecules “pack” more efficiently inside the ordered zeolite structure. In estimating the permeation of mixtures across zeolite membranes, both size and configurational entropy effects need to be properly accounted for. A model to estimate the permeation fluxes across zeolite membranes is developed by combining the Real Adsorbed Solution Theory to describe the mixture sorption, and the Maxwell-Stefan diffusion equations. The utility of the model is demonstrated by means of two case studies: (1) permeation of methane and n-butane, and (2) permeation of n-hexane and 2,2 dimethylbutane across a silicalite membrane.