Diffusive Mass Flux Research Articles

A predictive model for the permeability of gas mixtures in glassy polymers

The transport of gaseous mixtures in glassy polymers is analyzed by means of a thermodynamic based model, which is applied to describe the permeability of CO2/CH4 50/50 binary mixtures in various glassy polymeric membranes. The approach relies on the description of the solubility behavior of penetrant/polymer mixtures provided by the nonequilibrium thermodynamics for glassy polymers (NET-GP), and considers the gradient in penetrant chemical potential of each species as the actual driving force of the diffusive mass fluxes. Such an approach is specialized to dilute solutions conditions, as it is typically of interest for the transport of light gas (e.g. CO2, N2, CH4, O2) in glassy polymeric membranes; that allows for the simple and successful prediction of the gas permeability of gas mixtures based on single component transport data, with no additional parameters required.The NET-GP model is used in combination with an equation of state (lattice fluid theory by Sanchez and Lacombe) to obtain the solubility of pure and mixed gases at various pressures and compositions, as well as the thermodynamic factors accounting for the dependence of chemical potentials of the solutes on the concentrations of both penetrants. An exponential dependence on penetrant concentration is used to describe the mobility coefficient behavior, so that only two adjustable parameters are required for the pure penetrant case (infinite dilution mobility and plasticization factor). A simple but effective linear mixing rule is considered to describe transport in the binary mixture case, which does not introduce any additional adjustable parameter due to the presence of a second penetrating species. The comparison with permeation data of gas mixtures in different polymers shows the good predictive ability of the model.

Entropy generation minimization for isothermal crystallization processes with a generalized mass diffusion law

Entropy generation minimization for a class of isothermal crystallization processes with a generalized mass diffusion law is investigated in this paper. For the given total mass of crystals, the optimality condition corresponding to the MEG (Minimum Entropy Generation) of the process is obtained firstly, and special cases for both the linear [g∝Δ(μ)] and the diffusive [g∝Δ(c)] mass diffusion laws are further derived. The optimization results are also compared with the two different mass diffusion strategies of CKCC (Constant Key Component Concentration) and CMDF (Constant Mass Diffusion Flux) operations. The results indicate that for the case with the linear law, the entropy generation rate for the MEG of the process keeps constant, and the total entropy generations for the strategies of CKCC and CMDF operations are equal to each other; while for the case with the diffusive law, the entropy generation rate for the MEG of the process decreases with the time, and the strategy of CMDF is superior to that of CKCC. The results can provide important guidelines for optimal design and operation of the crystallization processes in engineering.

Non-Fickian mass transfer in swelling polymeric non-porous membranes

The aim of this contribution is to formulate a model that investigates the probable occurrence of non-Fickian behavior of mass transport in a viscoelastic polymeric non-porous membrane that swells. Using both continuum mechanics and non-equilibrium thermodynamics approaches, we derive a set of coupled time evolution equations that explicitly incorporate the stress-diffusion mutual interaction. In this 3-D macroscopic model, the trace Π (i.e. the polymer elastic osmotic pressure) and traceless [σp] parts of the polymer second order stress tensor σp serve as the structural independent state variables. In the one dimensional setting, scaling analysis leads to the emergence of three relevant groups of physical parameters namely the core and surface diffusion Deborah numbers De and Deb and a diffusion-stress coupling constant E0 whose effects are examined for both sorption and permeation unsteady processes. We have solved numerically the 1-D governing equations expressed in the material coordinates and obtained profiles for the solvent concentration, non-Fickian diffusion mass flux, stress distribution, swelling, mass-uptake and stress-uptake. The model is capable of predicting a wide range of non-Fickian behaviors, including the two-stage and overshoot cases; and the confrontation with experimental data selected from the literature shows a good agreement.

New phenomenological model of multicomponent gas diffusion

A new phenomenological model is proposed for diffusion of multicomponent gas mixtures. Obvious general structural relations between the mass diffusion fluxes and the gradients of gas mixture components are obtained within the framework of the model in the case of the multicomponent diffusion coefficients expressed in terms of the “intrinsic” diffusion coefficients of individualmixture components. The molecular mean free path model can be used to estimate the latter.

A new method for the transport mechanism coupling of shale gas slippage and diffusion

In view of the current status that different literature applies different coupling methods to the calculation of shale gas flow, and in order to clarify the relation between slippage and several diffusions, in this paper the slippage effect and various diffusions are analyzed first by theoretical analysis and mathematical models according to the definitions and the mechanisms of microscopic motions. Afterwards, allowing for the spatial effect of the adsorbed molecules on gas flow, the concept “wall-associated diffusion” is proposed for the first time to represent the gross effects of Knudsen diffusion and surface diffusion, and it is pointed out that wall-associated diffusion is equivalent to slippage effect. Therefore a new coupling way where wall-associated diffusion and slippage effect are replaceable and no superposition of them is needed in flow calculation, is proposed. The case study shows that when the capillary radius ranges from 5 nm to 2000 nm, the relative error between wall-associated diffusion and slippage effect mass flux is fairly small, namely less than 10% in the vast majority of the range. The difference between mean values of wall-associated diffusion and slippage effect mass flux in the whole aperture range is 1.4×10- 6 kg·m-2·s-1. That is, the relative error between the mean values is only 5.8%. Therefore, the new method satisfies the requirements for engineering calculations. Taking parameter selection, unfinished improvements in mathematical models of relevant mechanisms and other factors into account, there is some room for further promoting the verification of the proposed method. The development of wall-associated diffusion has practical significance and multiple research significance. And the new coupling way reveals the relation between slippage and diffusions, which prevents reduplicated superposition of shale gas flow mechanisms in nano-scale pores and can well change the status where the current coupling methods for shale gas flow are not consistent, thus specifying a new direction in the quantitative calculations for shale gas development.

Research on shale gas transportation and apparent permeability in nanopores

Shale gas transport mechanisms in matrix nanopores include the Knudsen diffusion, transition diffusion, slip flow of free gas and surface diffusion of adsorbed gas. Previous research focused on shale gas Knudsen diffusion, slip flow and surface diffusion. In this paper, we present a shale gas mass flux model for the shale formation matrix that incorporates transition diffusion, slip flow and surface diffusion. A set of shale formation data was used to analyse the sensitivity of the gas mass flux to the pore radius and pressure for each gas transport mechanism. The results show that each gas transport mechanism mass flux contribution varies with the pore radius and pressure. The results of the gas mass flux sensitivity analysis are as follows: (1) the slip flow mass flux increases with increasing pore radius and pore pressure, (2) the surface diffusion mass flux decreases with increasing pore pressure and pore radius, and (3) the transition diffusion mass flux decreases with increasing pore pressure and has only a slight dependence on the pore radius. Furthermore, the gas type notably influences the apparent permeability. We recommend this model for the description of the shale gas mass flux in matrix nanopores. It provides insights into the effects of transition diffusion, slip flow and surface diffusion on the shale gas transportation in nanopores.

A new combination of nanoparticles mass diffusion flux and slip mechanism approaches with electrostatic forces in a natural convective cavity flow

Understanding of the phenomena involved in the mixture of nanoparticles and fluid requires more investigation in terms of many aspects. Both diffusion process and slip mechanisms are studied here with a new approach applied in the governing equations in Mixture model. The new approach tested for laminar natural convective flow inside a cavity (with two differentially heated walls) by using ANSYS FLUENT 15.0 with the presence of Alumina and Zinc Oxide nanofluids. The new slip mechanism covers the effects of virtual mass, pressure gradient, lift, buoyancy, centrifugal, van der Waals attraction and electrical double layer repulsion forces. All the slip mechanism and source terms in the governing equations are implemented as User Define Functions in ANSYS FLUENT 15.0. The numerical results provided good agreement with experiments performed in this study. Depending on the volume fraction, heat transfer may improve or deteriorate, as reported by others. The comparison indicates that the ability of the proposed method is mainly associated to the concentration distribution, and of course in the ranges of volume fraction studied here. It is also found that the diffusion fluxes change the concentration profile near the diabatic walls, while the slip mechanism will be dominant in adiabatic walls.

Equilibrium and non-equilibrium concentration fluctuations in a critical binary mixture.

When a macroscopic concentration gradient is present across a binary mixture, long-ranged non-equilibrium concentration fluctuations (NCF) appear as a consequence of the coupling between the gradient and spontaneous equilibrium velocity fluctuations. Long-ranged equilibrium concentration fluctuations (ECF) may be also observed when the mixture is close to a critical point. Here we study the interplay between NCF and critical ECF in a near-critical mixture aniline/cyclohexane in the presence of a vertical concentration gradient. To this aim, we exploit a commercial optical microscope and a simple, custom-made, temperature-controlled cell to obtain simultaneous static and dynamic scattering information on the fluctuations. We first characterise the critical ECF at fixed temperature T above the upper critical solution temperature Tc, in the wide temperature range [Formula: see text] °C. In this range, we observe the expected critical scaling behaviour for both the scattering intensity and the mass diffusion coefficient and we determine the critical exponents [Formula: see text], [Formula: see text] and [Formula: see text], which are found in agreement with the 3D Ising values. We then study the system in the two-phase region (T < T c). In particular, we characterise the interplay between ECF and NCF when the mixture, initially at a temperature Ti, is rapidly brought to a temperature T f > T i. During the transient, a vertical diffusive mass flux is present that causes the onset of NCF, whose amplitude vanishes with time, as the flux goes to zero. We also study the time dependence of the equilibrium scattering intensity I eq, of the crossover wave vector q co and of the diffusion coefficient D during diffusion and find that all these quantities exhibit an exponential relaxation enslaved to the diffusive kinetics.

Rational design of mass diffusion metamaterial concentrators based on coordinate transformations

Recent advances in coordinate transformations of Fick's equation have paved the way for the design of metamaterial devices that can manipulate mass diffusion flux. The control of diffusion paths has a great potential for the design of novel catalytic and separation systems in chemical and biomolecular engineering. In order to explore these new applications, it is necessary to understand mass diffusion in coordinate transformation metamaterial devices. In this work, we present a comprehensive study on the impact of structure and material properties on the resultant physical properties of mass concentrator metamaterial shells. The concentration gradient at the core, the total mass flow rate towards the core, and the disturbance of the external concentration field are systematically examined in order to provide guidelines for the rational design and fabrication of metamaterial mass concentrators. A practical case is also presented where the concentration of oxygen diffusing in a polymeric system is studied.

Numerical investigation of radiative optically-dense transient magnetized reactive transport phenomena with cross diffusion, dissipation and wall mass flux effects

High temperature electromagnetic materials fabrication systems in chemical engineering require ever more sophisticated theoretical and computational models for describing multiple, simultaneous thermophysical effects. Motivated by this application, the present paper addresses transient magnetohydrodynamic heat and mass transfer in chemically-reacting fluid flow from an impulsively-started vertical perforated sheet. Thermal radiation flux, internal heat generation (heat source), Joule magnetic heating (Ohmic dissipation), thermo-diffusive and diffuso-thermal (i.e. cross-diffusion) effects and also viscous dissipation are incorporated in the mathematical model. To facilitate numerical solutions of the coupled, nonlinear boundary value problem, non-similar transformations are employed and the partial differential conservation equations are normalized into a dimensionless system of momentum, energy and concentration equations with associated boundary thermal conditions. An implicit finite difference method (FDM) is utilized to solve the unsteady equations. Verification of the FDM solutions for dimensionless velocity, temperature and concentration functions is achieved with a variational finite element method code (MAGNETO-FEM) and also a network simulation method code (MAG-PSPICE). The influence of the emerging thermo-physical parameters on transient velocity, temperature, concentration, wall shear stress, Nusselt number and Sherwood number is elaborated. The flow is accelerated with increasing thermal radiative flux, Eckert number, heat generation and Soret number whereas the flow is decelerated with greater wall suction, heat absorption, magnetic field and Prandtl number. Temperatures are also observed to be elevated with magnetic parameter, radiation heat transfer, Dufour number, heat generation (source) and Eckert number with the contrary effects computed for increasing suction parameter or Prandtl number. The species concentration is enhanced with Soret number and generative chemical reaction whereas it is depressed with greater wall suction, Schimidt number and destructive chemical reaction parameter

Representing effects of aqueous phase reactions in shallow cumuli in global models

AbstractAqueous phase reactions are important, sometimes dominant (e.g., for SO2), pathways for the oxidation of air pollutants at the local and/or global scale. In many current chemical transport models (CTMs), the transport and aqueous reactions of chemical species are treated as split processes, and the subgrid‐scale heterogeneity between cloudy and environmental air is not considered. Here using large eddy simulation (LES) with idealized aqueous reactions mimicking the oxidation of surface‐originated SO2 by H2O2 in shallow cumuli, we show that the eddy diffusivity mass flux (EDMF) approach with a bulk plume can represent those processes quite well when entrainment/detrainment rates and eddy diffusivity are diagnosed using a conservative thermodynamic variable such as total water content. The reason is that a typical aqueous reaction such as SO2 aqueous oxidation is relatively slow compared to the in‐cloud residence time of air parcels in shallow cumuli. As a result, the surface‐originated SO2 is well correlated with and behaves like conservative thermodynamic variables that also have sources at the surface. Experiments with various reaction rate constants and relative abundances of SO2 and H2O2 indicate that when the reaction timescale approaches the in‐cloud residence time of air parcels, the errors of the bulk plume approach start to increase. Treating chemical tracer transport and aqueous reaction as split processes leads to significant errors, especially when the reaction is fast compared to the in‐cloud residence time. Overall, the EDMF approach shows large improvement over the CTM‐like treatments in matching the LES results.

Real-time Simulation of Dual-Layer Catalytic Converters Based on the Internal Mass Transfer Coefficient Approach

A new numerically efficient simulator for dual layer catalysts is implemented and its performance is demonstrated using the example of the dual layer ammonia oxidation catalyst. For the solution of the radial mass balances, each washcoat layer is represented by a single volume element and the diffusive mass fluxes into and within the washcoat are computed based on the concept of the internal mass transfer coefficients. The performance of the new simulator is compared against a reference simulator that fully resolves the concentration profiles in the washcoat. For a steady state test case, the error introduced by the new solution scheme is below 2 % for all relevant exhaust components. For a transient test case, the deviations become more significant. For the simulation of a WHTC, the maximum deviation in the NH3 outlet concentration is 23.3 %, whereas the cumulated NH3 emissions deviate by 10.6 %.With the new simulator, the 1800 s of the transient WHTC cycle can be simulated within 53 s on a standard laptop, 30 times faster than real time.

Hydrogen sulfide formation, fate, and behavior in anhydrite-sealed carbonate gas reservoirs: A three-dimensional reactive mass transport modeling approach

A novel hydrogeochemical modeling approach is developed to unravel thermochemical sulfate reduction (TSR) in hydrocarbon reservoirs. Our numerical model couples a web of interconnected hydrogeochemical reactions to three-dimensional (3-D) and reservoir-wide diffusive mass transport. Our modeling approach simulates a semigeneric gas reservoir sealed by anhydrite. The calculated diagenetic processes fit the observations in reservoirs affected by TSR: formation of water, precipitation of calcite, metal (di-)sulfides, and elemental sulfur as replacements of dissolved anhydrite at the expense of CH4(g), as well as formation of hydrogen sulfide (H2S). By varying input parameters, the crucial factors controlling TSR have been identified. Our results highlight that reservoir-wide diffusive mass transport is one prerequisite for TSR. An increase in the rate constant of abiotic sulfate reduction (ASR) and in diffusive mass fluxes, as well as lack of precursor minerals for metal (di-)sulfide precipitation, can increase the souring intensity and accelerate H2S outgassing. In contrast, precipitation of elemental sulfur, which is stable according to the chemical thermodynamics, weakens H2S formation. Our modeling shows that TSR is complex and cannot be represented by the single reaction ASR and by simple correlations between the rate constant of ASR and the H2S gas content. The application of 3-D reactive transport modeling presented here, despite its semigeneric nature, provides a good example of how such an approach can be used ahead of drilling. Our modeling helps to investigate TSR in time and space to quantify the mass conversion of all reactants involved within this web and to predict the souring level.

Evaluation of PBL Parameterizations in WRF at Subkilometer Grid Spacings: Turbulence Statistics in the Dry Convective Boundary Layer

Abstract Planetary boundary layer (PBL) parameterizations in mesoscale models have been developed for horizontal resolutions that cannot resolve any turbulence in the PBL, and evaluation of these parameterizations has been focused on profiles of mean and parameterized flux. Meanwhile, the recent increase in computing power has been allowing numerical weather prediction (NWP) at horizontal grid spacings finer than 1 km, at which kilometer-scale large eddies in the convective PBL are partly resolvable. This study evaluates the performance of convective PBL parameterizations in the Weather Research and Forecasting (WRF) Model at subkilometer grid spacings. The evaluation focuses on resolved turbulence statistics, considering expectations for improvement in the resolved fields by using the fine meshes. The parameterizations include four nonlocal schemes—Yonsei University (YSU), asymmetric convective model 2 (ACM2), eddy diffusivity mass flux (EDMF), and total energy mass flux (TEMF)—and one local scheme, the Mellor–Yamada–Nakanishi–Niino (MYNN) level-2.5 model. Key findings are as follows: 1) None of the PBL schemes is scale-aware. Instead, each has its own best performing resolution in parameterizing subgrid-scale (SGS) vertical transport and resolving eddies, and the resolution appears to be different between heat and momentum. 2) All the selected schemes reproduce total vertical heat transport well, as resolved transport compensates differences of the parameterized SGS transport from the reference SGS transport. This interaction between the resolved and SGS parts is not found in momentum. 3) Those schemes that more accurately reproduce one feature (e.g., thermodynamic transport, momentum transport, energy spectrum, or probability density function of resolved vertical velocity) do not necessarily perform well for other aspects.

Flow and non-Fickian mass transport in immiscible blends of two rheologically different polymers

The aim of this paper is to derive a new mathematical model that investigates, under isothermal conditions, the coupling arising among non-Fickian diffusion, flow, and the deformation of the internal structure in a ternary mixture composed of a simple fluid and a binary immiscible blend of two rheologically different polymers. Along with the overall linear momentum density vector and the global scalar mass density, the diffusion mass flux density vector and the scalar mass fraction of the simple fluid, three symmetric second-rank tensor variables are adopted to track down the blend microstructural dynamic changes: two are contravariant polymer conformation tensors, and one is a covariant interface area tensor. The general equation for the nonequilibrium reversible and irreversible coupling (GENERIC) is used to derive, on the mesoscopic level of description, a set of coupled and nonlinear time evolution and constitutive equations for the selected state variables. As a special case, a new reduced model for the structure-controlled flow-free diffusion is derived, and the non-Fickian character of diffusion is shown. Using the method of characteristics, the paper also examines the physics of propagation of both linear dispersive and nonlinear hyperbolic waves produced by the disturbances in the simple fluid concentration. Explicit new formulas for the characteristic speed, phase velocity, and attenuation are derived.

A NUMERICAL STUDY OF A THREE-DIMENSIONAL PROTON EXCHANGE MEMBRANE FUEL CELL (PEMFC) WITH PARALLEL AND COUNTER FLOW GAS CHANNELS

Modeling the heat and mass transport in micro channel is being used extensively in researches and industrial applications. The aim is optimizing fuel cell designs before building a prototype for engineering application. In this study, numerical, three-dimensional, single phase computational fluid dynamics model of a proton exchange membrane fuel cell with both the gas distribution flow channels and the Membrane Electrode Assembly (MEA) has been developed. A single set of conservation equations which are valid for the flow channels, gas-diffusion electrodes, catalyst layers, and the membrane region, are solved by finite volume technique. The present simulated straight channels PEMFC model, accounts for major transport phenomena and the performance. Additionally, the effect of reversing the flow direction at cathode side has been investigated to study the fuel cell performance and species distribution. The results showed that, in the PEMFC with the counter flow channels, the output current density has been decreased and also the kind of species distributions has been influenced by this phenomenon. It is very important to model the back diffusion and electro-osmotic mass flux for determination of ionic conductivity of membrane which affects the performance of fuel cell. Finally, the numerical results are validated by available experimental data.

Diffusion in Mixtures of Reacting Thermoelastic Solids

A chemically reacting mixture of elastic solids is considered. As a constitutive assumption, the peculiar functions (such as the free energy, the entropy, and the stress) of a constituent are taken to be functions of a set of variables pertaining to that constituent. The interaction terms, namely the growth of mass, linear momentum, and energy, are allowed to depend on the set of variables pertaining to all of the constituents. While the dependence on the mass density is usually disregarded, the paper accounts also for such a dependence, which seems to be in order especially in connection with reacting mixtures where the mass densities change also in the reference configuration. The thermodynamic restrictions are derived by starting from the non-negative value of the sum of entropy growths and involving the properties of the peculiar functions. The results so obtained for stresses and chemical potentials are examined in connection with similar schemes (swelling solids). While the correct relations for the mass diffusion flux arise from balance equations, an analysis is given of whether and how Fick-type models are acceptable possibly depending on the fluid or solid character of the mixture.

Unsteady three-dimensional stagnation-point flow and heat transfer of a nanofluid with thermophoresis and Brownian motion effects

An unsteady three-dimensional stagnation-point flow of a nanofluid past a circular cylinder with sinusoidal radius variation is investigated numerically. By introducing new similarity transformations for the velocity, temperature, and nanoparticle volume fraction, the basic equations governing the flow and heat and mass transfer are reduced to highly nonlinear ordinary differential equations. The resulting nonlinear system is solved numerically by the fourth-order Runge–Kutta method with the shooting technique. The thermophoresis and Brownian motion effects occur in the transport equations. The velocity, temperature, and nanoparticle concentration profiles are analyzed with respect to the involved parameters of interest, namely, unsteadiness parameter, Brownian motion parameter, thermophoresis parameter, Prandtl number, and Lewis number. Numerical values of the friction coefficient, diffusion mass flux, and heat flux are computed. It is found that the friction coefficient and heat transfer rate increase with increasing unsteadiness parameter (the highest heat transfer rate at the surface occurs if the thermophoresis and Brownian motion effects are absent) and decrease with increasing both thermophoresis and Brownian motion parameters. The present results are found to be in good agreement with previously published results.

A numerical study of fixed bed reactor modelling for steam methane reforming process

AbstractA numerical comparison of the pseudo‐homogeneous, conventional heterogeneous, and simplified heterogeneous reactor models is performed for the steam methane reforming process. The pseudo‐homogeneous reactor model consists of a set of partial differential equations, where the diffusional limitations are accounted for by considering the effectiveness factor. The heterogeneous model is divided into two categories: conventional and simplified heterogeneous models. In the conventional heterogeneous model, there are separate equations for the fluid phase species mass balance, the fluid inside the catalyst pores. This is type of reactor model is needed when there is a considerable amount of inter‐phase mass transfer resistance present in the process. However, in the simplified heterogeneous reactor model the mass transport phenomenon is accounted for by accounting the efficiency factors. The model is validated against literature data. Several closures for the intra‐particle mass diffusion fluxes, the Maxwell–Stefan, Wilke, dusty gas, and Wilke–Bosanquet models, have been compared on the level of the catalyst pellet and the impacts of the different particle flux closures on the reactor performance are investigated.The simulations show that the conventional heterogeneous reactor model is necessary for the SMR process, because the effectiveness factor values of different reactions of the SMR process vary along the reactor axis. The maximum deviation between the pseudo‐homogeneous and the conventional heterogeneous reactor model is less than 38 %, whereas between the conventional and simplified heterogeneous reactor models it is less than 21 %. A parametric study of the transport phenomena on the pellet level is recommended prior to any large‐scale reactor simulation to determine what the rate determining transport mechanisms are.

Salt Diffusion Through a Bentonite-Polymer Composite

AbstractBentonites are commonly used as chemical containment barriers to minimize liquid flow and contaminant transport. However, chemicals can adversely affect bentonite performance to the extent that modified bentonites have been developed to improve chemical resistance relative to traditional (unmodified) bentonites. The present study focused on the diffusion of potassium chloride (KCl) through a bentonite-polymer composite, or BPC, that was known to behave as a semipermeable membrane. Specifically, the effective diffusion coefficients, D*, for chloride (Cl−) and potassium (K+) were measured and correlated with previously measured membrane efficiency coefficients, ω, for the BPC. The values of D* at steady-state for chloride (Dss,Cl−*\$\end{document}) and potassium (Dss,K+*\$\end{document}) decreased as the ω values increased. The decrease in Dss,Cl−*\$\end{document} and Dss,K+*\$\end{document} was approximately a linear function of (1 − ω), which is consistent with previous research performed on unmodified Na-bentonite contained within a geosynthetic clay liner (GCL). In contrast to the previous GCL tests, however, Dss,Cl−*\$\end{document} values for the BPC generally were greater than the Dss,K+*\$\end{document} values, and the differences between Dss,Cl−*\$\end{document} and Dss,K+*\$\end{document} decreased as KCl concentration increased. The apparent discrepancy between Dss,Cl−*\$\end{document} and Dss,K+*\$\end{document} is consistent with excess sodium (Na+) in the BPC prior to testing and the requirement for electroneutrality during testing. Also, despite an apparent linear trend in diffusive mass flux for K+, lack of agreement between the ratio of the diffusive mass flux of K+ relative to that for Cl− as required on the basis of electroneutrality at steady state suggested that steadystate diffusive mass flux for K+ had probably not been achieved due to continual K+-for-Na+ cation exchange. Nonetheless, the excess Na+ and bentonite modification did not affect the fundamental correlation between D* and ω, which requires that D* approaches zero as ω approaches unity (D* → 0 as ω → 1).