Maxwell-Stefan Approach Research Articles

A Stefan-Maxwell Model of Single Pore Pressurization for Langmuir Adsorption of Gas Mixtures

A model based on the application of the Maxwell-Stefan approach has been used to describe the dynamics of intraparticle transport (pore diffusion, surface diffusion and convection) in a single pore during and after a pressurization process. The model was first compared with the model proposed by Taqvi and Levan (Adsorption, 2, 299–309 (1996)) for a linear adsorption isotherm. The effect of several parameters (pressurization rate, adsorption capacity, bulk gas-phase mole fraction, adsorption affinity and pore radius) was studied, evaluating the relative importance of each mass-transport mechanism in different conditions. A binary mixture of an inert and an adsorbable component was considered first, extending the analysis of the pore radius effect to a ternary mixture. In general, surface diffusion is dominant with very low pore radius, whereas gas-phase fluxes dominate in a large pore. However, depending on the value of the bulk gas-phase mole fraction (which is related to the surface coverage level through the adsorption equilibrium isotherm), the equilibrium and rate parameters, and the surface to volume ratio, surface diffusion cannot be always neglected for large pores. More generally, system non-linearity can switch the dominant mechanism and create fronts.

The generalized Maxwell–Stefan model for diffusion in zeolites:: sorbate molecules with different saturation loadings

Using the Maxwell–Stefan approach, expressions have been derived for the diffusion of mixtures of hydrocarbons in zeolites wherein the individual components have different saturation loadings. This Maxwell–Stefan diffusion model, in combination with the Ideal Adsorbed Solution (IAS) theory and the single-component adsorption isotherms, provides a superior, qualitative and quantitative, prediction of the permeation fluxes of ethane/methane and propane/methane mixtures through a silicalite-1 membrane. The difference in adsorption saturation loadings becomes apparent in the entropy effects in the mixture adsorption.

Modeling of Diffusion in Zeolites

Diffusion of adsorbed molecules in zeolites plays an important role in the use of zeolites as adsorbents in separation processes and in shape-selective catalysis. Computational chemistry is in a stage where diffusion phenomena, even for multicomponent diffusion, can be treated at a level of high accuracy. The present review presents most of the results on diffusion in zeolites obtained by classical Molecular Dynamics, dynamic Monte-Carlo approaches, Transition-State Theory, and the Maxwell-Stefan approach. Reactive and non-reactive conditions are considered.

Dynamic Modelling of Reactive Absorption with the Maxwell-Stefan Approach

Modelling and design of reactive absorption are based on the theoretical description of the reaction and mass transport in multicomponent systems. The multicomponent nature of these phenomena leads to complex process behaviour due to the superposition of many driving forces multicomponent diffusion, chemical interactions, convective flows, multicomponent thermodynamic interplay, etc. For this reason, adequate theoretical description of multicomponent reactive systems calls for the application of the Maxwell-Stefan equations and, further, for the use of coupled mass transfer equations together with the relevant reaction kinetics. On this basis, a two-phase, gas-liquid reactive system is considered and a general dynamic model is developed for its design. Both the film and bulk reaction mechanisms are allowed for. This dynamic rate-based approach leads to a system of partial differential equations, which have to be discretized in the axial direction. The resulting DAE system is solved numerically. As an application example, the reactive absorption of sour gases in an air purification process with packed columns is simulated. For this case, an additional account of the electrical potential gradient is involved because of the presence of electrolytes. Simulation results are presented for the H 2 S scrubber with three liquid distributors and a structured packing section. For the validation of the model, pilot plant steady state experiments were carried out at Thyssen Still Otto in Duisburg, Germany. The simulation results are in good agreement with the experimental data.

Application of the Maxwell–Stefan theory to the transport in ion-selective membranes used in the chloralkali electrolysis process

The results of a fundamental mass transport model based on the Maxwell–Stefan approach are compared to experimental data obtained by Akzo-Nobel for a Dupont Nafion ion-selective membrane as used in chloralkali electrolysis processes. The main problem in the application of the Maxwell Stefan based mass transfer model to the chloralkali electrolysis process is a lack of available diffusivities for the membrane. Estimation of these diffusivities in the membrane based on a method presented by Wesselingh et al. (1995. Chem. Engng J., 57, 75–89) gave unrealistic high membrane potential drops. Therefore, another method was followed. First, a sensitivity analysis was carried out which resulted in a reduced set consisting of the dominating Maxwell–Stefan diffusivities. First estimates of these remaining diffusivities were determined for single layer sulfonic and a carboxylic membranes. With a slight adjustment of the values of the diffusivities obtained for the separate sulfonic and carboxylic layers, the performance parameters of the DuPont Nafion membrane could be predicted well for a reference experiment. These diffusivities also proved to be suitable for other anolyte strengths. However, for other catholyte strengths and current densities these diffusivities (even after a correction for the water uptake according to the method of Wesselingh et al. (1995. Chem. Engng. 5., 57, 75–89)) did not result in a good agreement between the simulated and experimentally observed performance parameters. Only after a correction of the diffusivities the simulations yielded approximately the same performance parameters as experimentally observed. From this it can be concluded that although a fundamental model is used in order to describe the mass transfer in a membrane, a single set of diffusivities is not sufficient in order to obtain the experimentally observed performance parameters at different process conditions. At this moment there is not enough knowledge on the exact phenomena taking place in the membrane in order to predict the necessary corrections of the diffusivities a priori. As long as there are no theoretically founded and reliable relations available to predict the Maxwell–Stefan diffusivities in a membrane (or accurate experimental data for these diffusivities) only a semi-empirical method as used in this study can serve as a basis for a further progress in the development of an existing (in this case DuPont Nafion) membrane.

Binary protein adsorption on gel‐composite ion‐exchange media

AbstractThe adsorption kinetics of mixtures of lysozyme and cytochrome‐C on S‐HyperD‐M, a cation exchanger composed of a rigid macroporous silica matrix whose pores are filled with a functionalized polyacrylamide gel, was characterized. Single‐component isotherms were found to follow approximately the steric mass‐action law of Brooks and Cramer, and an extension of this model to a binary system was consistent with two‐component uptake equilibrium results. Transient adsorption experiments were done to investigate the kinetics of sequential and simultaneous adsorption of the two proteins. At low protein concentrations, adsorption is essentially noncompetitive, and mass transfer is controlled by the external film resistance. At higher concentrations, equilibrium becomes competitive and coupling of the intraparticle diffusion fluxes occurs. A model based on a Maxwell–Stefan approach, where the driving force for diffusion is expressed in terms of the chemical potential gradient, was in approximate agreement with the experimental results.

A Maxwell-Stefan approach to modelling the cross-flow ultrafiltration of protein solutions in tubular membranes

A non-parameterised model is developed to predict the crossflow membrane ultrafiltration permeate flux for protein solutions. A relatively simple hydrodynamic model is used to predict the rate of growth of the concentration polarisation boundary layer along the length of a tubular membrane. Protein transport within the boundary layer is evaluated using the Maxwell–Stefan transport equations, accounting for electrostatic interactions created by filtering charged protein molecules. Numeric simulations were performed with and without considering viscous interactions in the concentration polarisation boundary layer. In the main, the model appears to qualitatively describe the ultrafiltration process very well when tested for a range of different operating parameters and solution conditions. However, at high TMPs viscous interactions were exaggerated and the model underpredicted permeate flux. Under such conditions, use of the model without accounting for viscous interactions seemed more appropriate. The model predictions are tested against sample experimental data for the protein BSA and the results were highly encouraging.

Maxwell-Stefan approach coupled drop population model for the dynamic simulation of liquid-liquid extraction pulsed column

Zimmermann and al. (1995) have proposed a model for the simulation of a multi-component extraction process based on Maxwell-Stefan approach and incorporating a drop population model. In the continuation of this work, this paper presents some recent developments, notably the extension to the dynamic and some improvements at the hydrodynamic level. The model is characterised by the coupling of two main aspects in separation processes: hydrodynamics and multi-component mass transfer. Hydrodynamics is described by a drop population model: fundamental mechanisms like transport, axial back-mixing and forward-mixing, drop breakage and interdrop coalescence, including Marangoni effect, are described in a detailed way. Multi-component mass transfer is computed using the Maxwell-Stefan approach. The proposed model is able to predict the component concentration profiles in the dispersed and continuous phases, hold-up and drop size distribution throughout the column and the interactions between mass transfer and hydrodynamics under operating conditions up to flooding.

Maxwell–Stefan analysis of multicomponent transient diffusion in a capillary and adsorption of hydrocarbons in activated carbon particle

This paper reports the results of detailed studies of the combined effect of the viscous flow and diffusive flow in the transient diffusion of a mixture containing N species in a capillary, and in the transient adsorption of a mixture of hydrocarbons in an activated carbon particle. The Maxwell–Stefan approach is applied to describe the interaction of diffusing species in the large pore network, where the viscous flow mechanism operates in conjunction with the other diffusion mechanisms. In the pure diffusion systems, the viscous flow affects significantly the overall diffusion flux when the molecular weights of the diffusing species are significantly different or when the pore size is small. In the adsorption systems, the viscous flow can enhance the adsorption process as the loss of mass from the gas phase may render the viscous flow significant. We test the theory with the adsorption of nitrogen/ethane/propane onto activated carbon, and it is shown that the theory describes well the adsorption kinetics of this mixture.

Effectiveness Factor for a Non-Isothermal Simple Catalytic Reaction with Combined Transport Processes: Maxwell-Stefan Approach

A two-step procedure is developed and tested for the prediction of effectiveness factors for stoichiometrically simple catalytic reactions. In the first step the dependences of state variables (concentrations, temperature, pressure) on the concentration of the key reaction component is obtained. The knowledge of these dependences allows then all the dependent variables in the catalyst pellet mass and heat balances to be expressed by this concentration. In the second step, a single balance differential equation is solved and the effectiveness factor is obtained. In addition, a deeper insight into the concentration and temperature conditions inside the pores is gained. Information of this kind can cast light on some features of the catalytic reactor behaviour. The analysis of ammonia synthesis illustrates this point.

Maxwell-Stefan approach in extractor design

A new mathematical model for the simulation of multicomponent liquid-liquid extraction columns using the rate-based approach is presented. The model is characterized by the coupling of two main aspects in separation processes: a sophisticated description of hydrodynamics and rigorous simulation of mass transfer. For mass transfer, the mass conservation equations for the continuous phase and each drop class of the dispersed phase are written using the Maxwell—Stefan equations for multicomponents mass transfer, taking into account the thermodynamic noidealities of liquid—liquid systems. For hydrodynamics, the model takes into account backmixing characteristics of the dispersed and the continuous phase as well as the basic mechanisms of convective transport, axial mixing, drop breakage and coalescence. This is handled by using a population balance model and the drop size distribution over the height of the column.

A film model based approach for simulation of multicomponent reactive separation

A general approach to the design of multicomponent mass transfer accompanied by chemical reactions is presented. Both homogeneous and heterogeneous reactions in liquid phase, as well as their combination, have been studied and incorporated into the film model. Because of the multicomponent nature of reactive separation, the mass transfer equations in the fluid phases are based on the Maxwell-Stefan approach. These equations are coupled by diffusional interactions in the gas (vapour) phase and by both diffusional and homogeneous chemical interactions in the liquid phase. Heterogeneous reaction mechanism has been accounted for by mass balance relationships. The solution obtained allows the determination of concentration profiles of the components along the contacting apparatus.

Multicomponent tray efficiencies accounting for entrainment

In multicomponent distillation, the component point efficiencies are highly sensitive to the composition profiles and are known to exhibit bizarre behavior. A major limitation of these efficiencies is the fact that they cannot be related to tray efficiencies for dispersed and plug flow of liquid, unless multicomponent effects are ignored — these would still pose a problem if the component point efficiencies assume large negative values. The Maxwell—Stefan approach to diffusion in multicomponent mixtures leads to a component point efficiency matrix which does not suffer from these drawbacks. In this work, rigorous methods are pesented for obtaining a tray efficiency matrix from a point efficiency matrix for dispersed flow and plug flow of a liquid phase, and accounting for the entrainment. The results of the simulation of the fractionation of two ideal and two non-ideal mixtures indicated the superiority of the matrix approach over the use of component efficiencies.

Multicomponent mass and energy transport on different length scales in a packed reactive distillation column for heterogeneously catalysed fuel ether production

A reactive distillation column was operated which was packed with acid ion exchange rings catalysing the production of the octane enhancer methyl-tert.-butylether (MTBE) from methanol (MeOH) and isobutene (IB). Two types of catalytic rings were used: a block polymer (BP) with a capacity of 4.54 meq(H+)/g and a glass supported precipitated polymer (GPP) with a capacity of 0.9 meq(H+)/g. Fluid dynamic experiments with pure MeOH and pure MTBE show that the catalytic packings, which are located in the upper column section, work under trickle flow conditions. Nonreactive distillation experiments reveal sustained macroscale oscillations of fluxes, pressure and temperature that are caused by the distillation behaviour of the binary mixture MeOH/IB. These oscillations were also detected in the reactive distillation process with the highly stable, but less active GPP catalyst. The local compositions and temperatures from the nonoscillatory reactive distillation experiments with the BP catalyst are subjected to a detailed analysis of microscale solid-liquid and vapour-liquid transport resistances based on the Generalised Maxwell-Stefan approach. It turns out that the mass transport resistances inside the catalyst can vary significantly along the packing. The main mass transport resistance at the vapour-liquid interface is located on the liquid side.

Multicomponent mass and energy transport on different length scales in a packed reactive distillation column for heterogeneously catalysed fuel ether production

A reactive distillation column was operated which was packed with acid ion exchange rings catalysing the production of the octane enhancer methyl- tert.-butylether (MTBE) from methanol (MeOH) and isobutene (IB). Two types of catalytic rings were used: a block polymer (BP) with a capacity of 4.54 meq(H +)/g and a glass supported precipitated polymer (GPP) with a capacity of 0.9 meq(H +)/g. Fluid dynamic experiments with pure MeOH and pure MTBE show that the catalytic packings, which are located in the upper column section, work under trickle flow conditions. Nonreactive distillation experiments reveal sustained macroscale oscillations of fluxes, pressure and temperature that are caused by the distillation behaviour of the binary mixture MeOH/IB. These oscillations were also detected in the reactive distillation process with the highly stable, but less active GPP catalyst. The local compositions and temperatures from the nonoscillatory reactive distillation experiments with the BP catalyst are subjected to a detailed analysis of microscale solid-liquid and vapour-liquid transport resistances based on the Generalised Maxwell-Stefan approach. It turns out that the mass transport resistances inside the catalyst can vary significantly along the packing. The main mass transport resistance at the vapour-liquid interface is located on the liquid side.