Geometrical relationships for ternary gas diffusion balances and criteria for multicomponent phenomena

Gas mixtures of more than two components are often encountered in chemical engineering. If the species are coupled by a diffusive process in the molecular diffusion regime, special multi-component phenomena may arise. Until now, however, there is only fragmentary information in literature on this topic. It is demonstrated how to derive criteria for multicomponent phenomena—for a whole set of flux couplings—based upon a simple geometric procedure. Osmotic and reverse diffusion in ternary systems now can be predicted in a simple and comprehensive manner. It is shown that both may also occur in ternary systems with coupling of diffusion and reaction.The proposed method uses diffusion balances, which permit an insight into concentration relationships along the diffusion path as a whole, without need for solution of the differential equations of molar fluxes.Linear balances are introduced as an additional multicomponent feature. Diffusion balance diagrams are presented as a mean to represent all possible boundary values on both ends of the diffusion path in the case of a diffusion barrier.     

Prediction of diffusion in a ternary solvent–solvent–polymer blend by means of binary diffusion data: Comparison of experimental data and simulative results

Multicomponent diffusion of solvents in polymeric systems is not completely understood, despite many scientific contributions to the topic. Literature scarcely offers measurement data on diffusion for model validation in such systems. In this work, the ternary systems consisting of poly(vinyl acetate) and the solvents toluene and methanol was investigated experimentally and numerically. By means of inverse micro Raman spectroscopy (IMRS) concentration gradients in drying thin films have been measured. Initial composition of the samples has been varied systematically in order to detect mutual influence of the solvents' diffusive behavior. It was shown that the mobility of the different species is increased in the presence of other solvents as predicted by theory. This experimental data is provided for model validation. A new expression to calculate the diffusion coefficients in ternary mixtures is proposed which only requires binary data. This expression is tested by means of a model‐based simulation to predict the drying of ternary polymer solutions in terms of concentration profiles and residual solvent content. The results are in very good agreement with the experiments. Cross terms diffusion coefficients and thermodynamic factors were not found to be necessary for a satisfying prediction. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016 , 133, 43899.     

Modeling of multicomponent mass diffusion in porous spherical pellets: Application to steam methane reforming and methanol synthesis

The main purpose of this paper is the derivation and evaluation of various diffusion flux models. For this aim, a comprehensive catalyst pellet problem has been simulated for two test cases: the steam methane reforming (SMR) and the methanol synthesis, as these two important chemical processes cover various aspects of a chemical reaction. The pressure, temperature, total concentration, species composition, viscous flow, mass and heat fluxes within the porous spherical pellet are included in the transient pellet model. Mass diffusion fluxes are described according to the rigorous Maxell–Stefan and dusty gas models, and the respectively simpler Wilke and Wilke–Bosanquet models. Simulations are performed with these fluxes defined according to both the molar averaged and mass averaged definitions. For the mass based pellet equations, a consistent set of equations is obtained holding only the mass averaged velocity. On the other hand, the closed set of molar based pellet equations hold both the molar averaged and mass averaged velocities as the fundamental energy balance and the momentum balance (Darcy law) are derived according to the mass averaged velocity definition, whereas the diffusion fluxes are defined relative to the molar averaged velocity. Identical results of the molar and mass based pellet equations were not obtained; however, the deviations are small. It is anticipated that these discrepancies are due to some unspecified numerical inaccuracies. However, efficiency factors have been computed for both processes and the values obtained compare well with the available literature data. Furthermore, efficiency factor sensitivity on parameters like pore diameter, tortuosity, temperature and pressure have been accomplished, and the classical simplifications of the pellet equations have been elucidated: isothermal condition, constant pressure, and neglecting viscous flow. The following conclusions are established for the reactor operating conditions used in the present work. The methanol synthesis: The simulation results of the methanol synthesis indicate that the classical assumptions are very fair for this process. Moreover, both Wilke and Wilke–Bosanquet models are good replacements for the more rigorous Maxwell–Stefan and dusty gas models. However, the simulation results are affected by Knudsen diffusion, thus the diffusion flux is most appropriately described by the Wilke–Bosanquet model. The SMR process: Knudsen diffusion hardly influences the results of the highly intraparticle diffusion limited SMR process. As the Wilke model does not necessarily conserve mass, we recommend the Maxwell–Stefan model because the simpler Wilke closure deviates with several percents. However, it is not elucidated whether these deviations are numerical problems arising from the large gradients of this process, or related to the choice of diffusion model. Isothermal and isobaric conditions can be assumed within the particle, but significant external temperature gradients are observed. Convective fluxes are much less than the diffusive fluxes, hence viscous flow can be neglected.     

Estimating the density and molar volume of ferrite-based ternary molten slags by geometrical model

Ferrite-based slags are the basic systems in the pyrometallurgical production of copper, and the density and molar volume of these systems are of vital importance for optimizing the metallurgical process. However, due to the high melting points of melts and the expensive trial costs, experimental methods are inefficient to obtain all the data needed. As a result, a geometrical model has been applied to the estimation of density and molar volume of ferrite-based ternary slags. The calculated results agree well with experimental data in three ferrite-based ternary systems of Fe₂O₃–CaO–FeO, Fe₂O₃–BaO–FeO and Fe₂O₃–CaO–Cu₂O. The mean deviation of the predicted results by the present model is lower than those by various empirical models for corresponding systems. The application of the present will enrich the data of density and molar volume for more ternary melts.     

STABILITY OF THE FILM MODEL FOR MULTICOMPONENT MASS TRANSFER

The computational algorithm for calculating rates of mass transfer in multicomponent mixtures using the film model proposed by Krishna and Standart can be unstable for certain physical processes. Successful prediction of the molar fluxes can usually be achieved by evaluating the diffusive fluxes at the appropriate position on the diffusion path.     

甲醇-乙酸-丙酸多元体系汽液平衡

用新型泵式沸点仪测定了在100kPa下甲醇-乙酸、甲醇-丙酸、乙酸-丙酸3个二元体系以及甲醇-乙酸-丙酸三元系在不同液相组成时的沸点,并用间接法T-p-x(温度、压力和液相摩尔分数)推算了3个二元体系的汽相平衡组成。3个二元体系活度系数分别用Wilson模型、NRTL模型、Margules模型和van Laar模型进行关联,用最小二乘法求出了它们的液相活度系数模型参数,同时,用这些模型参数来计算它们的汽相摩尔分数。所得的液相活度系数来计算3个二元体系的过量吉布斯自由能函数,且所研究的所有体系中各组分之间不存在共沸点。用3个二元体系Wilson模型参数对所测的三元体系数据进行关联,建立该系统汽液平衡的热力学模型并计算平衡时的汽相摩尔分数和泡点温度。由面积积分法检验这些模型参数计算的3个二元体系相平衡数据,得到很好的热力学一致性。     

STABILITY OF THE FILM MODEL FOR MULTICOMPONENT MASS TRANSFER

The computational algorithm for calculating rates of mass transfer in multicomponent mixtures using the film model proposed by Krishna and Standart can be unstable for certain physical processes. Successful prediction of the molar fluxes can usually be achieved by evaluating the diffusive fluxes at the appropriate position on the diffusion path.     

New models for tracer diffusion coefficients of hard sphere and real systems: Application to gases, liquids and supercritical fluids

In this work very accurate expressions for the tracer diffusion coefficient of hard sphere (HS) and real systems are proposed. The new HS model depends explicitly on the reduced density of solvent, and on the ratios of the diameters and masses of solute and solvent. It provides a very good representation of molecular dynamics data taken from literature: average absolute relative deviation, AARD = 4.44%. With respect to real fluids, the proposed model was developed according to Rice and Gray approach, and is based on the previous HS equation. The model involves only one parameter and requires temperature, solvent density, and solute and solvent molecular weight and LJ force constants (these are estimated as function of the critical temperature and molar volume). Results calculated for 309 systems and 5341 data points gave rise to AARD = 4.26%, and shows the model interprets equally well the diffusive phenomena of gases, liquids and SCFs.     

Ternary fluid lattice Boltzmann simulation of dynamic interfacial tension induced by mixing inside microdroplets

A ternary fluid color-gradient lattice Boltzmann model is proposed to investigate the effect of mixing-induced dynamic interfacial tension on the diffusive mixing of two fluids inside microdroplets moving in a third continuous phase through a baffled channel. The diffusion coefficient of binary mixtures and dynamic interfacial tension in this model can be directly defined and independently adjusted. The simulation results show that the dynamic interfacial tension between miscible binary fluids at interfaces with ambient phase would lead to the motion of the interface and redistribution of solutes inside the droplet during mixing. The larger initial interfacial tension gradient would improve mixing efficiency at early stages by promoting faster solute flow. The present model can be easily applied to quantitatively characterize the mixing behavior inside droplets in the practical processes involving the dynamic interfacial tension phenomenon and inspire new designs for mixing intensification.     

EXPERIMENTAL STUDIES OF DIFFUSION FLUXES F TERNARY DISTILLATION OF THE ACETONE-METHANOL-ETHANOL YSTEMS BY A WETTED-WALL OLUMN

Measurements of mass and diffusion fluxes for ternary distillation of the acetone-methanol-ethanol systems were made by a wetted-wall column under total reflux conditions for wide ranges of the liquid concentrations. Intermediate component methanol showed peculiar behavior of a finite mass flux near the zero-driving force region. This was explained as being due to the effect of the interfacial velocity caused by mass fluxes of the other components. The diffusion fluxes were shown to be proportional to concentration driving forces, whereas the mass fluxes were not. Discussions on the use of the number of transfer units for correlation of mass transfer data in ternary distillation were also made.     

多物理场中热扩散效应对传热传质的影响

为了便于消除或控制室内挥发性有机化合物(VOCs),分析了VOCs在室内环境中的扩散规律。基于不可逆过程热力学原理,在考虑交叉耦合扩散效应的基础上,建立了3个物理量的梯度驱动下封闭空间内自然对流传热传质的数学模型。采用数值方法研究了室内环境中同时存在温度梯度、湿度梯度和VOCs浓度梯度时的自然对流传热传质现象,并着重考察了多物理量的梯度耦合作用下热附加扩散效应对传热传质的影响,展示了流场、温度场和浓度场等热附加扩散准则数STC的变化状况,研究结果表明,多物理场中热附加扩散效应对传热传质具有一定的影响,且具有复杂多解性。     

Measurement of concentration profiles using confocal Raman spectroscopy in multicomponent polymeric coatings—model validation

Concentrations of the solvents were measured using confocal laser Raman Spectroscopy for two ternary systems, poly(styrene)—tetrahydrofuran—p‐xylene and poly(methyl methacrylate)—ethylbenzene—tetrahydrofuran, during drying at room temperature. The concentrations were compared with predictions of drying models, which utilize several existing theories for mutual diffusion coefficients for polymer solvent systems. Of the nine free volume parameters required to predict diffusion coefficients of binary systems, four for each of the four pairs studied here were estimated as suggested by the literature. Estimation was done by minimizing the difference between predictions of the model and experimental weight loss data for each binary pair. It is found that the predictions of the models which include cross term diffusion coefficients are in better agreement with measured concentrations than those which ignore the cross terms. © 2012 Wiley Periodicals, Inc. J. Appl. Polym. Sci., 2013     

Diffusion in gels

The diffusion of solutes in gels is comprehensively reviewed. Because it has been a source of confusion, precise definitions of the gel diffusion coefficient are presented and discussed. Theories as to the effect of the gel substance on the course of diffusion are critically evaluated. These include the obstruction effect, hydrodynamic drag and other frictional couplings, alteration of solvent properties and (for homogeneous gels) the free volume theory. A large proportion of the data on diffusion in gels to be found in the literature is displayed, with the exception of those systems where binding of the solute is a major factor. The success of the theories in accounting for these results is examined. It is concluded that for heterogeneous gels the obstruction effect is prevalent, for organic solvent-polymer systems the free volume theory has had some success while diffusion of both macromolecules and micromolecular solutes in homogeneous gels is not well understood and deserves more experimental effort.     

蔗糖三相流态化结晶过程中离集与混合特性的研究

在一总高６５０ｍｍ，直径６０ｍｍ的三相流化床实验装置上，分别采用压降法和取样法测定了不同操作条件下二元混合物蔗糖晶体颗粒浓度沿轴向的分布，并建立了表征颗粒混合与分级程度的数学模型，得到不同条件下的混合系数。结果表明，混合系数即混合程度随气速增加而增大，但随液速增加略有减小；随大颗粒比率的增加，床中的离集程度有所增大。     

节能型变换催化剂曲折因子和气体有效扩散

设计自制改进的Wicke-Kallenbach扩散池,按稳态法实验测定了常压313K下N2-CO,N2-CO2,CO-CO2二组分气体和CO+CO2-N2,CO+N2-CO2,N2+CO2-CO三组分气体在还原态LB型节能高温变换催化剂内的逆流扩散通量,实验所得二组分和三组分气体扩散通量之比符合Graham气体扩散规律,表明扩散池设计合理,测定过程稳定、数据可靠。同时说明气体在还原态LB型节能高温变换催化剂内的扩散是过渡区扩散。根据平行交联孔模型,计算出以平均孔半径为基准的曲折因子和以孔分布为基准的曲折因子以及各气体的有效扩散系数。     

One‐dimensional Model of Oxygen Transport Impedance Accounting for Convection Perpendicular to the Electrode

A one‐dimensional (1D) model of oxygen transport in the diffusion media of proton exchange membrane fuel cells (PEMFC) is presented, which considers convection perpendicular to the electrode in addition to diffusion. The resulting analytical expression of the convecto‐diffusive impedance is obtained using a convection–diffusion equation instead of a diffusion equation in the case of classical Warburg impedance. The main hypothesis of the model is that the convective flux is generated by the evacuation of water produced at the cathode which flows through the porous media in vapor phase. This allows the expression of the convective flux velocity as a function of the current density and of the water transport coefficient α (the fraction of water being evacuated at the cathode outlet). The resulting 1D oxygen transport impedance neglects processes occurring in the direction parallel to the electrode that could have a significant impact on the cell impedance, like gas consumption or concentration oscillations induced by the measuring signal. However, it enables us to estimate the impact of convection perpendicular to the electrode on PEMFC impedance spectra and to determine in which conditions the approximation of a purely diffusive oxygen transport is valid. Experimental observations confirm the numerical results.     

Bounds for effective diffusivities using moment and transport constraints

Upper and lower bounds for an effective diffusivity characteristic of binary diffusion through a heterogeneous medium are calculated by imposing geometric and transport constraints on the pore size distribution. The method relies upon identifying moment spaces induced by Tchebycheff systems closely related to diffusion experiments and a priori information about the geometry. If statistics beyond the porosity and average pore size are included, temperature dependent tortuosity factors result from the strong coupling between the diffusive flux within a capillary and the pore size distribution. It is shown that hybrid statistical information, such as the porosity and a single diffusion experiment, can delimit significantly the allowed region for extrapolated data.     

Diffusion pathways of benzene, toluene and p-xylene in MFI

The diffusion of alkyl-substituted aromatic molecules in H-ZSM-5 was investigated by means of the frequency response method decoupling particle size effects and intracrystalline diffusion. For zeolite crystals above 5 μm average diameter, the transport in the zeolite pores exerts a significant effect on the overall transport causing anisotropic diffusion as the aspect ratio of the aromatic molecules increases. Diffusion of benzene is nearly isotropic, while p-xylene shows marked differences between the diffusive processes in the straight and sinusoidal channel system of ZSM-5. The isotropic diffusion of benzene is rationalized on the basis of its ability to reorient between the two channel systems without major hindrances. For p-xylene, switching between the channels is only possible by energetically unfavorable rotational motions leading to a low probability for changing between both channel systems.     

TERNARY EXTRACTION MODELING USING THE REDLICH-KISTER EXPANSION

Mathematical models for liquid-liquid extraction processes are not widely available. The correlation of liquid-liquid equilibrium for such a model is often difficult. One correlation, based upon the Redlich-Kister expansion for the molar excess Gibb's free energy, has been shown to accurately represent ternary liquid-liquid equilibrium systems used in liquid-liquid extraction operations.

The use of this correlation in an extraction model was investigated and found to be suitable for a ternary liquid-liquid extraction model with only minor limitations. A successful ternary liquid-liquid extraction model was developed using this equilibrium correlation.