Maxwell-Stefan Model Research Articles

A transition in diffusion behaviors of organic liquid mixtures in dense polymer membranes

Multi-component mass transfer frameworks, such as the Maxwell-Stefan approach, parameterized with the pure component diffusion and sorption behaviors of different substances in polymer materials have been used in predicting complex mixture permeation through polymer membranes. However, a change in the physical state of a polymer membrane (e.g., swelling) or the polymer dynamics (e.g., plasticization) is likely when the membrane is exposed to a high activity organic solvent mixture with strong chemical affinity for the polymer. These changes complicate the accurate estimation of diffusion coefficients for mixtures of solvents based on knowledge of pure component diffusivities within the polymer. Here, we present 120 organic solvent reverse osmosis (OSRO) permeation test results for hydrocarbon liquid mixtures with varying compositions in three different polymers and compare the experimental results to the predictions of a Flory-Huggins coupled Maxwell-Stefan model with two different diffusion modalities. The effect of the chemical affinity between the mixture and the membrane – estimated by the Hansen solubility difference – on the diffusion behaviors of individual solvents permeating through the membrane was investigated via the usage of this model. Our work demonstrates that polymer membranes undergo a reduction for the effectiveness of diffusion-based separation near a specific threshold level of solubility difference (8 MPa0.5) from the mixture and that this difference is consistent across several different membrane materials.

Mathematical modeling of bioactive extraction from cashew apple: Maxwell-Stefan approach to resolve effect of internal diffusivity and external mass transfer

Every year 2.2 giga-tons of cashew apples are produced and 80 % (unutilized fruit) of them are wasted and contain useful bioactives such as polyphenols and tannins (high molecular weight polyphenols). These bioactives can be separated using extraction: the nutraceutical value of which is of the order of 1 billion USD/year. However, extraction kinetics are normally fitted to simple phenomenological models. These models fail to establish the effect of shrinkage, temperature, thermodynamic parameters, and internal resistances on diffusion of solute. Hence, the objective of this study was to develop a Maxwell-Stefan model that can overcome the above shortcomings and optimize the extraction process of bioactives from cashew apples. The optimized parameters for the extraction process obtained were − solid:solvent ratio- 1:6, particle size (of solid) 125–250 μm, and the extraction temperature as 30 °C. The non-random two liquid (NRTL) activity coefficient model was first used to capture the thermodynamics of extraction effectively, followed by its utilization in the development of Maxwell-Stefan model to resolve the effect of internal diffusivity and external mass transfer coefficient. The model gave a good fit (R2 > 0.90) for all the parameters and could thus predict the extraction data greatly for all operating parameters including temperature, particle size, and solid:solvent ratio. This makes it eminently suitable for scale-up studies.

On the validation and applicability of multiphysics models for hydrogen SOFC

Solid oxide fuel cells (SOFC) are a viable alternative for environmentally-friendly conversion of hydrogen into energy and multiphysics simulation can be used to diminish the experimental effort to improve their efficiency. However, an appropriate model of the involved processes and their parameters must be chosen. This paper studies the effects of choice between Maxwell–Stefan and Fick’s law models, and uncertainty of electrode ionic conductivity σion and anodic reference exchange current density i0,ref,f, on cell performance as implemented in the COMSOL Multiphysics® software. In the case of Maxwell–Stefan, peak average power output increased by 21.9% as σion varies from 10−3 to 10−1 S/cm, while the model based on Fick’s law shows an increase of 55.2%. The Maxwell–Stefan model exhibits an increase in peak power of 6% as i0,ref,f ranges from 0.4 to 0.8 A/cm2, and the Fick’s law model an increase of 8.2%. The dependence of the Maxwell–Stefan model on σion is characterized as logarithmic in the studied range. The Maxwell–Stefan model is deemed preferable because its lower sensitivity to the studied parameters helps mitigate uncertainty. It is concluded that despite its limitations, multiphysics modeling is a useful tool for directing research on SOFC materials owing to its descriptive potential.

An energy-stable and conservative numerical method for multicomponent Maxwell–Stefan model with rock compressibility

Numerical simulation of gas flow in porous media is becoming increasingly attractive due to its importance in shale and natural gas production and carbon dioxide sequestration. In this paper, taking molar densities as the primary unknowns rather than the pressure and molar fractions, we propose an alternative formulation of multicomponent Maxwell–Stefan (MS) model with rock compressibility. Benefiting from the definitions of gas and solid free energies, this MS formulation has a distinct feature that it follows an energy dissipation law, and namely, it is consistent with the second law of thermodynamics. Additionally, the formulation obeys the famous Onsager's reciprocal principle. An efficient energy-stable numerical scheme is constructed using the stabilized energy factorization approach for the Helmholtz free energy density and certain carefully designed formulations involving explicit and implicit mixed treatments for the coupling between molar densities, pressure, and porosity. We rigorously prove that the scheme inherits the energy dissipation law at the discrete level. The fully discrete scheme has the ability to ensure the mass conservation law for each component as well as preserve the Onsager's reciprocal principle. Numerical tests are conducted to verify our theories, and in particular, to demonstrate the good performance of the proposed scheme in energy stability and mass conservation as expected from our theories.

Improved numerical methods for simulating complex mixture transport across asymmetric polymer membranes using a Maxwell–Stefan model

Improved numerical methods are presented for predicting the total flux and permeate composition of complex hydrocarbon mixtures permeating through asymmetric glassy polymer membranes. The proposed methods are comprehensively compared against a suite of existing methods for several challenging test cases; namely, three and nine component mixtures permeating through a glassy polymer, SBAD-1, and a five component mixture permeating through a glassy polymer, PIM-1. We find that the proposed shooting algorithm is the most accurate, efficient, and robust for these test cases. In comparison, existing methods based on common model simplifications suffered from large errors and frequent convergence failures for higher component mixtures, while those based on numerical discretization techniques suffered from higher cost and needed much more careful tuning to provide reliable solutions.

Finite volumes for the Stefan–Maxwell cross-diffusion system

Abstract The aim of this work is to propose a provably convergent finite volume scheme for the so-called Stefan–Maxwell model, which describes the evolution of the composition of a multi-component mixture and reads as a cross-diffusion system. The scheme proposed here relies on a two-point flux approximation, and preserves at the discrete level some fundamental theoretical properties of the continuous models, namely the non-negativity of the solutions, the conservation of mass and the preservation of the volume-filling constraints. In addition, the scheme satisfies a discrete entropy–entropy dissipation relation, very close to the relation that holds at the continuous level. In this article, we present this scheme together with its numerical analysis, and finally illustrate its behaviour with some numerical results.

Mixed-matrix ZIF-loaded membranes for effective separation of bio-butanol from fermentation broth

Mixed matrix composite membranes offer superior performance for separation of alcohols from fermentation broth. In this study bio-butanol (Butanol) is recovered from Acetone: Butanol: Ethanol (ABE) fermentation broth using ZIF-71 -PDMS-PVDF membranes. The significance of this work is the supported PVDF layer that was prepared using a dual coagulation bath to increase the surface roughness and contact angle. The addition of hydrophobic ZIF-71 particles into the matrix increases their thermal stability, hydrophobicity, and strength of the membrane. The synthesized membranes exhibited higher hydrophobic characteristic with a contact angle of around 120°. The Pervaporation efficiency of these membranes was analyzed by various experimental parameters such as ZIF loading, permeate pressure, feed temperature, and stirring speed. The membrane interaction parameters were calculated using Flory Huggins and Flory-Rehner equations. The membrane performance was also assessed by Maxwell-Stefan model fitting along with the evaluation of the friction coefficient. The butanol flux (50 g/m2h) was estimated using mathematical modeling and compared with the experimental results (120 g/m2h). The studies inferred that the models are well-fitted for butanol extraction which implies the penetrant-penetrant interaction is lower than penetrant-membrane interactions. A comparison of the present study with literature is well-defined to scale-up the process at an industrial level.

Modeling and Optimizing Anode Catalyst Layer for Direct Ammonia Fuel Cell

Great progress has been made in recent years in the development of low-temperature direct ammonia fuel cells (DAFCs), motivated by the recognition that ammonia is a carbon-free hydrogen carrier with high energy density, low production cost, and ease in liquefaction at ambient temperature. However, the sluggish kinetics of ammonia electrooxidation and especially complicated mass transport in the anode catalyst layer hinder the further development of DAFCs. In this work, a three-dimensional two-phase multicomponent DAFC model considering the effect of ammonia crossover has been developed. Maxwell-Stefan model, Darcy's law and Brinkman equation are utilized to simulate the multicomponent fluid motion and transport. The predicted polarization curve simulates all experimental results well. The model shows the rate of ammonia crossover decreases with the increase of current density. Besides, the effects of the physicochemical property of the anode catalyst layer, including porosity, thickness and PtIr loading, on cell performance are investigated. The modeling results indicate that decreasing porosity and increasing thickness can slightly improve the electrochemical performance of DAFCs at high current density. Meanwhile, higher PtIr loading can effectively reduce voltage loss before approaching the limiting current density.

Modeling of Diffusive Transport of Polymers Moments Using Limiting Cases of the Maxwell–Stefan Model

AbstractA polymer distribution is usually represented by its moments. Thus, to calculate transport in a polymer system, a formulation for the transport of moments of the polymer is needed. This is only possible if the moments close or if there is a suitable closing condition. To archive this, two simplifications of the Stefan–Maxwell diffusion are derived, which convert the transport equation of polymeric species to a closed set of transport equations for the polymer moments. The first approach corresponds to an infinitely diluted polymer system, whereas the second one describes a highly concentrated polymer system. Both formulations are compared with the full Stefan‐Maxwell model of a ternary mixture of a solvent and two polymer species of different chain length.

A single-phase diffusion model for gas injection in tight oil reservoirs

This paper presents a mechanistic simulation study of one-dimensional multicomponent diffusion when the miscible injectant diffuses into a tight porous medium through the fracture/matrix interface with constant pore volume and temperature. The numerical implementation of diffusion based on the dusty gas model uses the fugacity gradient for each component in the mixture as the driving force to the diffusive flux. The Peng-Robinson equation of state is used to model the non-ideal interactions among components in the miscible diffusive process. Phase stability analysis by minimization of the Helmholtz free energy is performed for each grid block at every time step to ensure that mixtures are single-phase fluids throughout the simulation.The main novelty of this paper lies in the diffusion model and its theoretical analysis, in which the fluid non-ideality affects the multicomponent diffusion through two pathways: the fugacity coefficients and the volume change on mixing that causes local pressures to change under ultra-low permeability in tight porous media. Previous studies based on the Maxwell-Stefan model did not consider the latter pathway, while others based on Fick's law are even more simplistic by not considering the non-ideal chemical potential.Analysis in this research showed that the Maxwell-Stefan model was inconsistent with its own assumption of no pressure gradient when non-ideal mixing was considered for tight reservoirs. The dusty gas model does not have this issue because it allows for pressure gradients to drive mass transfer by Knudsen diffusion. The non-ideal interaction of components should be properly characterized and utilized to enhance the early-time flux through the fracture/matrix interface in miscible gas injection into a tight reservoir. Case studies show that the volume change on mixing may substantially increase local pressures and the rate of mass transfer in tight reservoirs. Also, the fugacity coefficients of oil components at infinite dilution in the solvent had a major influence on the rate of diffusion. These two factors highlight the importance of properly characterizing reservoir fluids through an equation of state.

A Comparison between Klinkenberg and Maxwell-Stefan Formulations to Model Tight Condensate Formations

Summary Cyclic solvent (gas) injection is an efficient recovery method for condensate reservoirs. However, in tight, unconventional formations, the added complexity of low permeability results in more physics at play, beyond the widely used Darcy model for conventional reservoirs. In this work, a rigorous mass transfer model is implemented considering the real gas version of the Maxwell-Stefan formulation to evaluate cyclic injection schemes in tight condensate reservoirs. This model is then compared to the more widespread used Klinkenberg formulation, which does not include molecular diffusion. An evaluation is performed to check if a simplified formulation can be used to provide reasonable results in modeling production and enhanced recovery in tight condensate formations. Verification of the implemented equations is performed using experiments (Maxwell-Stefan model) and a commercial reservoir simulator (Klinkenberg model). Furthermore, the cell length used for the numerical studies is selected based on a sensitivity study to evaluate how numerical dispersion impacts recovery factor and liquid saturation for different cell sizes. By comparing the Klinkenberg model with different tangential momentum accommodation coefficient (TMAC) values to the Maxwell-Stefan model during primary production, it is possible to select a value of TMAC that can match closely the recovery values of lighter components when using the Maxwell-Stefan equations. However, for heavier hydrocarbon fractions, difference in recovery is more accentuated owing to increased molecule size (more molecular friction). This results in differences in condensate yield during primary production that may be relevant in a field scale. In the cyclic injection scheme, the importance of accounting for frictional effects between molecules is demonstrated using the Maxwell-Stefan formulation. In this case, molecular diffusion fluxes are influenced by high composition gradients. This results in differences between the Maxwell-Stefan and Klinkenberg models in terms of gas stored and hydrocarbon produced during cyclic injection simulations. Furthermore, a sensitivity study on operational parameters in the cyclic injection stage demonstrated that increasing the length of production cycles may be more beneficial than increasing the length of injection or soaking cycles. For the simulations in this study, the gas is injected above the dewpoint and pressure diffusivity is at least one order of magnitude higher than the other physics present in the process. Therefore, increasing the length of production cycles allows for recovery of heavier hydrocarbon fractions that remain in the gas phase. In this work, it is demonstrated that using a rigorous mass transfer formulation, such as the Maxwell-Stefan equations, can provide more information on a per component basis when evaluating cyclic injection schemes in tight condensate reservoirs.

A discrete boundedness-by-entropy method for finite-volume approximations of cross-diffusion systems

Abstract An implicit Euler finite-volume scheme for general cross-diffusion systems with volume-filling constraints is proposed and analyzed. The diffusion matrix may be nonsymmetric and not positive semidefinite, but the diffusion system is assumed to possess a formal gradient-flow structure that yields $L^\infty $ bounds on the continuous level. Examples include the Maxwell–Stefan systems for gas mixtures, tumor-growth models and systems for the fabrication of thin-film solar cells. The proposed numerical scheme preserves the structure of the continuous equations, namely the entropy dissipation inequality as well as the non-negativity of the concentrations and the volume-filling constraints. The discrete entropy structure is a consequence of a new vector-valued discrete chain rule. The existence of discrete solutions, their positivity, and the convergence of the scheme is proved. The numerical scheme is implemented for a one-dimensional Maxwell–Stefan model and a two-dimensional thin-film solar cell system. It is illustrated that the convergence rate in space is of order two and the discrete relative entropy decays exponentially.

A Numerical Study of Hydrogen Production via High-temperature and Low-temperature Water-Gas Shift Reactors’ System: The Multi-Scale Modeling Approach and Simulation

The primary purpose of this study is to develop an advanced, and comprehensive multi-scale mathematical models of a packed bed reactors (PBRs) carrying out high and low temperature water-gas-shift reactions (WGSRs) for the hydrogen production. In industrial hydrogen generation applications, the water-gas-shift reactors are considered at high (called HTSR) and low (called LTSR) temperature stages with a cooling process between them. Therefore, detailed and advanced numerical studies on the HTSR and the LTSR in series are carried out to assess the overall performance of hydrogen production system. After completing a single-pellet, non-isothermal, steady-state simulation, we couple our model with a non-isothermal (adiabatic), steady-state packed-bed reactor model to form a hybrid multi-scale reactor model. The velocity, temperature and species’ concentration profiles along both the reactor length and the pellet radius are captured by using rigorously defined momentum, energy, and species transport models, accounting for the physical mechanisms involved in the system such as convection, conduction, and reaction-diffusion. The model’s equations are simultaneously solved for each domain: bulk gas domain and catalyst-pellet domain. The rigorous Maxwell-Stefan Model is applied on the reactor scale to account mass diffusion fluxes. On the other hand, Dusty Gas Model is considered to describe mass diffusion fluxes for the single pellet scale. Studies that include a broad range of the operating conditions and design parameters are carried out in this paper, in order to investigate the upper and lower limit conditions’ effects on the results.

Analysis of mass diffusion theory and models for high-temperature multi-component gases

The correct knowledge of physical theory and models of the mass diffusion for high-temperature multi-component gas flows is of importance for accurately simulating the aerothermodynamic properties of hypersonic vehicles. Mass diffusion mechanism from the phenomenological theory (the Fick’s and the modified Fick’s model) and the kinetic theory (the self-consistent effective binary diffusion model and the Stefan-Maxwell model) is theoretically analyzed to explore the reasons for model differences in application to CFD simulations of Earth reentry flows. Theoretical analysis shows that the other three models rather than the Fick’s model ensure the self-consistent condition. A difference term Ai is proposed to represent the discrepancy of the self-consistent effective binary diffusion model and the modified Fick’s model. Numerical results indicate that whether the mass diffusion model satisfying the self-consistent condition has great influence on surface heat transfer, with the maximum difference up to 18% under the fully-catalytic wall. The aerothermodynamic properties obtained by the modified Fick’s model and the self-consistent effective binary diffusion model are in good agreement, which can be explained by the small Ai values. It is proved that theoretical derivations and numerical results are consistent to conclude that the modified Fick’s model and the self-consistent effective binary diffusion model of rigorous physical fundamental are suitable to describe the mass diffusion process of high-temperature multi-component gases.

Mathematical approaches to modelling the mass transfer process in solid oxide fuel cell anode

In the recent literature, the influence of lean fuel, type of fuel, velocity profile and the flow direction on the performance operation of solid oxide cells (SOCs) is commonly investigated. SOCs generate electricity when they operate in fuel cell mode (SOFC – Solid Oxide Fuel Cell) or produce hydrogen in electrolysis mode (SOEC – Solid Oxide Electrolyzer Cell). They consist of several layers, including two porous electrodes separated by a gas-tight electrolyte. Typical anode porosity (fuel electrode in SOFC mode) varies from 20% to 50% (after sintering). The porosity value may limit the mass transport and the overall performance of SOC. This work investigates computationally and experimentally, the existing challenge such as diffusion mechanism in porous medium using the example of SOC electrode. A set of mass transport models (Fick's laws, modified Fick's law, Maxwell-Stefan Model, Dusty Gas Model) are described and discussed, highlighting their similarities, differences, advantages, and disadvantages as well as restrictions in usage. Finally, a detailed computational fluid dynamics (CFD) model of the diffusive transport in porous electrode was elaborated and validated. An experimental campaign was conducted using anode supported SOC, fabricated at the Institute of Power Engineering in Poland. Simulations were performed of the mass transport of pure hydrogen and pure water vapor or carbon dioxide (substitute of real product of electrochemical reaction in SOC for validation purposes) through porous anode. Tests were conducted in the wide flow range 10 [ml/min] up to 200 [ml/min] for both gases. The results demonstrate that hydrogen mole fractions reached similar values (0.5–0.6) at both outlets only for the maximum residence time or high porosity of anode. This implies that residence time and gas diffusion can be controlled by a combination of operating and microstructural parameters and can be estimated by numerical simulation using computational fluid dynamics. The main motivation and outcome of this work were to reduce the cost and time required for the optimization of the mass transport process through SOC's porous electrode.

Energy stable modeling of two-phase flow in porous media with fluid–fluid friction force using a Maxwell–Stefan–Darcy approach

Two-phase incompressible flow in porous media plays an important role in various fields including subsurface flow and oil reservoir engineering. Due to the interaction between two phases flowing through the pores, the fluid–fluid friction force may have a significant effect on each phase velocity. In this paper, we propose an energy stable (thermodynamically consistent) Maxwell–Stefan–Darcy model for two-phase flow in porous media, which accounts for the fluid–fluid friction. Different from the classical models of two-phase flow in porous media, the proposed model uses the free energy to characterize the capillarity effect. This allows us to employ the Maxwell–Stefan model to describe the relationships between the driving forces and the friction forces. The driving forces include the pressure gradient and chemical potential gradients, while both fluid–solid and fluid–fluid friction forces are taken into consideration. Thermodynamical consistency is the other interesting merit of the proposed model; that is, it satisfies an energy dissipation law and also obeys the famous Onsager's reciprocal principle. A linear semi-implicit numerical method is also developed to simulate the model. Numerical simulation results are provided to show that the fluid–fluid friction force can improve the oil recovery substantially during the oil displacement process.

Predictive study of succinic acid transport via nanofiltration membranes using a Maxwell–Stefan approach

AbstractBACKGROUNDTransport of organic acids through nanofiltration membranes varies considerably under different conditions. In the present work, a Maxwell–Stefan approach coupled to the Henderson–Hasselbach equation was applied to describe succinic acid permeation through nanofiltration membranes in binary and ternary solutions. Permeate flux and solute rejection were evaluated under several test conditions, by modifying variables such as pH, gauge pressure, succinic acid concentration and presence of other solute (glucose, as a neutral solute).RESULTSThe model performed well in describing all experimental results for succinic acid/water binary solutions, except for pH values close to the isoelectric point of the membranes, due to electroviscous effect changes. It was also noticed that other effects, such as solvation, are important and it is necessary to include them into the model for ternary solutions.CONCLUSIONSThe Maxwell–Stefan model was able to describe the succinic acid binary solution nanofiltration performance. © 2020 Society of Chemical Industry (SCI)

Maxwell-Stefan modeling and experimental study on the ionic resistance of cation-selective membranes in concentrated lye solutions

Both experimental investigation and mathematical modeling have been combined to clarify the influence of membrane properties, temperature, electrolyte concentration, and current density on membrane resistance of Nafion 117 in concentrated lye solutions. The ionic resistance was measured with and without membrane using four electrodes for 15 wt% and 32 wt% sodium hydroxide, temperatures up to 90 °C, and current densities up to 25 kA/m2. The results from the measurement using Direct Current (DC) method as well as Electrochemical Impedance Spectroscopy (EIS) method indicate that membrane resistance is a function of temperature and lye concentration but is independent of current density. A mathematical model based on the Maxwell-Stefan approach has been developed to predict the ionic membrane resistance, and the model has been validated using the measured experimental data. A more suitable semi-empirical correlation for Maxwell-Stefan diffusivities is proposed by replacing the expressions for binary diffusivities based on infinite dilution with the concentration-dependent binary diffusivities. The new proposed correlation performs better in the model validation with the experimental data than the expressions using infinite dilution diffusivities.

A Three-Dimensional Microstructure-Scale Simulation of a Solid Oxide Fuel Cell Anode—The Analysis of Stack Performance Enhancement After a Long-Term Operation

In this research, we investigate the connection between an observed enhancement in solid oxide fuel cell stack performance and the evolution of the microstructure of its electrodes. A three dimensional, numerical model is applied to predict the porous ceramic-metal electrode performance on the basis of microstructure morphology. The model features a non-continuous computational domain based on the digital reconstruction obtained using focused ion beam scanning electron microscopy (FIB-SEM) electron nanotomography. The Butler–Volmer equation is used to compute the charge transfer at reaction sites, which are modeled as distinct locally distributed features of the microstructure. Specific material properties are accounted for using interpolated experimental data from the open literature. Mass transport is modeled using the extended Stefan–Maxwell model, which accounts for both the binary, and the Knudsen diffusion phenomena. The simulations are in good agreement with the experimental data, correctly predicting a decrease in total losses for the observed microstructure evolution. The research supports the hypothesis that the performance enhancement was caused by a systematic change in microstructure morphology.

Study on the Separation of H2 from CO2 Using a ZIF-8 Membrane by Molecular Simulation and Maxwell-Stefan Model.

The purification of H2-rich streams using membranes represents an important separation process, particularly important in the viewpoint of pre-combustion CO2 capture. In this study, the separation of H2 from a mixture containing H2 and CO2 using a zeolitic imidazolate framework (ZIF)-8 membrane is proposed from a theoretical point of view. For this purpose, the adsorption and diffusion coefficients of H2 and CO2 were considered by molecular simulation. The adsorption of these gases followed the Langmuir model, and the diffusion coefficient of H2 was much higher than that of CO2. Then, using the Maxwell–Stefan model, the H2 and CO2 permeances and H2/CO2 permselectivities in the H2–CO2 mixtures were evaluated. Despite the fact that adsorption of CO2 was higher than H2, owing to the simultaneous interference of adsorption and diffusion processes in the membrane, H2 permeation was more pronounced than CO2. The modeling results showed that, for a ZIF-8 membrane, the H2/CO2 permselectivity for the H2–CO2 binary mixture 80/20 ranges between 28 and 32 at ambient temperature.