The length scale that is usually considered in reactive transport calculations, i. As a consequence, this approximation must be correct. Therefore, the Poisson equation reduces to a local electro-neutrality condition, and there is no transient local charge build-up: J c is thus constant throughout the sample.

After discretization on a grid cell, Equation 48 becomes:. Combining Equations 50 and 39 , with J c being constant throughout the sample, yields:. J c,d x k is a known quantity from Equation 40 and the value of J c can thus be computed. The implementation of these equations in PHREEQC made it possible to unravel some of the diffusion properties of cementitious materials. In particular, the role of the diffuse layer and its average charge in the establishment of the measured current and ion breakthrough during electro-migration experiments was quantified, demonstrating that reactive transport can be used as an essential tool to understand and quantify highly coupled transport processes in nanoporous media Appelo It is usually assumed that early extracts closely resemble the in situ composition of the pore water.

Reactive transport modeling of advective displacement experiments have been carried out in a handful of studies with the consideration of the presence of a diffuse layer and the dual continuum model described in a previous section Grambow et al. In all of these studies, a simplifying approximation was made: the porosity affected by the advective flow was set to the same exact value as the initial bulk porosity defined with the dual continuum model Fig.

While a significant portion of the diffuse layer porosity may remain indeed unaffected by the advective flow, it is unlikely that this portion is always equal to that of the diffuse layer porosity, which must be seen itself as a convenient simplified representation of the sample microstructure see previous section and Fig.

This displacement should result in the appearance of coupled flux because of charge imbalance Fig. In this last case, the parameters must be seen as being apparent rather than intrinsic to the material studied. Advective displacement studies can be seen as the adaptation to intact clayey rocks of earlier experiments carried out on compacted bentonite or montmorillonite samples McKelvey and Milne ; Kemper and Maasland ; Milne et al.

Ultrafiltration processes were clearly evidenced with steady state output solutions having a lower electrolyte concentration than input solutions. It is thus not possible to model these experiments with the model shown in Fig. The influence of the diffuse layer on water and ion water transport processes in nanoporous media is most often taken into account implicitly using empirical phenomenological coefficients Soler ; Malusis et al. As shown in a previous section, this modeling approach is restricted to systems in which a simple electrolyte such as NaCl is considered.

In the absence of thermal or electrical potential gradients, the ion flux of a single salt is modeled as the result of four contributions Soler ; Sun et al. Chemical osmosis corresponds to the viscous flow of water down the water activity gradient or up the osmotic pressure gradient , which drags solutes along. Neglecting the water compressibility, the following equation is obtained for the osmotic pressure gradient:. The hyperfiltration flux corresponds to a correction to the advective and chemical osmosis flux due to the exclusion of solutes from the diffuse layer:.

The diffusion coefficient is usually taken at the same value for anions and cations salt diffusion coefficient and corrected from the membrane efficiency coefficient. It follows:. Equation 59 or similar equations have been widely used in the literature to characterize membrane properties of clay materials Soler ; Malusis and Shackelford , ; Malusis et al. The coefficient of membrane efficiency is however dependent on the electrolyte nature and concentration, and as noted by Malusis and Shackelford , the development of concentration gradients within the membrane preclude the calculation of intrinsic w values from experimental data using Equation The no current condition imposes:.

The first term of the numerator of the right part in Equation 62 corresponds to a streaming potential, while the second term of the right part corresponds to a diffusion potential. Equation 62 is then reinserted into Equation 60 :. Equation 63 can be further rearranged to yield:.

Equation 65 makes clear that, in multicomponent systems, the coefficient of osmotic efficiency is specific to each chemical species. In this last case, the porous media has no membrane properties. Equation 64 has not been implemented in a reactive transport code yet. To do this, it would be necessary to relate the linear fluid velocity to hydrostatic and osmotic pressure terms in a multi-component model framework. If the entire porosity is affected by the advective fluid flow, the coefficients of osmotic efficiency can be calculated on the basis of the average pore concentrations computed in the previous section.

Similarly, the osmotic pressure gradient can be calculated with Equation 56 combined with one of the models used to estimate the concentration in a charged pore. However, the concentration used for calculating the osmotic pressure may be different from the average concentration in the pore. It would be surprising in any case if the advective flow affects uniformly the entire porosity, especially in clayey rocks and cementitious materials that exhibit a large range of pore sizes from nanometer size interlayer porosity to few micrometers macropores Curtis et al.

It may be thus necessary to subdivide the porosity into different domains, in which specific surface charge and fluid flow are distributed heterogeneously. Multiple-porosity model for fluid flow in shale reservoirs are currently developed to capture the effect of the complexity of the shales pore structure on fluid flow Yan et al. The consideration of ionic transport in modeling work adds one more level of complexity because of the presence of charged surfaces that make it necessary to adopt a multi-continuum approach.

The current implementations in reactive transport codes of coupled transport processes in the presence of a diffuse layer are in agreement with the theory of irreversible thermodynamics processes. The multi-component capability of these codes makes it possible to overcome many limitations encountered with simplified approaches that are limited to single salt transport. In particular, the transport of electrolyte background ions and tracer ions can be simulated simultaneously, demonstrating the existence of highly coupled processes during their transport through nanoporous media.

In addition, reactive transport codes are not limited to the simulation of systems with fixed surface charge and fixed diffuse layer dimensions, and can incorporate microstructural information in dual- to multi-continuum descriptions of the porosity. Applications of these capabilities to the simulation of clayey and cementitious material properties highlighted the prominent role of the diffuse layer in the transport properties of ions. This review provides some insights to advance reactive transport modeling in the direction of a full implementation of multi-component advective flux in the presence of a diffuse layer.

In particular, the osmotic efficiency parameter must be revisited so that ion and neutral molecule specific properties can be simulated simultaneously. Holistic models of flow and ion transport in nanoporous charged media still face significant but ultimately energizing scientific and computational challenges.

Carl I. We gratefully acknowledge, Thomas Gimmi and Josep Soler for their constructive review of the manuscript. Examples of layered phases in clayey and cementitious materials. Elsevier, p 5—31, Copyright , with permission from Elsevier. C—S—H layer structure according to Grangeon et al AFm structure according to Marty et al. A : 3-D visualization of the microstructure of an illite sample compacted at 1. C: Pore-width distribution obtained from a combination of bulk and microscopic characterization techniques. A : Water and ion distribution at the vicinity of a montmorillonite surface according to MD simulations.

B: Conceptual model of EDL structure. Journal of Colloid and Interface Science — Copyright , with the permission of Elsevier. A , B and C correspond to an ionic strength of 0. No aqueous complexation in bulk solution was considered for this calculation. Calculations were done at K. Difference of average diffuse layer concentration values computed with the Poisson—Boltzmann c PB equation and the mean electrostatic potential c MEP model Eqns.

Difference of average diffuse layer concentration values computed with the Poisson—Boltzmann c PB equation and the Dual continuum model c Dual model described in Figure 6 and Equations 23 and Same caption as in Fig. A: experimental setup. The clay sample 1 is in contact with a low salinity reservoir 2 and high salinity reservoir 3 by liquid lines. Advective displacement experimental setup. Advective flow in the presence of a diffuse layer and related problematics of charge balance. A: The advective flow is limited to the bulk water porosity. An electro-neutral solution displaces an electro-neutral solution and the charge balance is maintained throughout the sample.

B: The advective flow displaces the bulk water porosity as well as a portion of the diffuse layer porosity. An electro-neutral solution displaces a charged solution: coupled processes must occur to maintain the local charge balance in the sample porosity and in the output solution. Example of experimental and modeling results obtained with an advective displacement experiment using a Callovian—Oxfordian argillite sample from the underground research laboratory in Bure Meuse-Haute Marne, France. The type of model used in the study corresponds to the model shown in Fig.

Terminology of coupled fluxes after de Marsily Sign In or Create an Account. User Tools. Sign In. Advanced Search. Article Navigation. Research Article September 01, Google Scholar. Steefel Carl I. CISteefel lbl. Reviews in Mineralogy and Geochemistry 85 1 : Article history first online:. Molecular diffusion in porous media is usually described in terms of Fick's First Law:. In the absence of an external electric field, there is no electrical current and so:.

The mobility of ion i in the porosity is:. J c,d is the current contribution from the chemical activity gradient:. Rearranging Equation 41 yields:. The corresponding chemical osmosis flux is:. In the presence of advective flow, the Nernst—Planck equation becomes:. Figure 1. View large Download slide. Table 1. Volume 85, Number 1. Previous Article Next Article. Benchmark reactive transport simulations of a column experiment in compacted bentonite with multispecies diffusion and explicit treatment of electrostatic effects.

Search ADS. Incorporating electrical double layers into reactive-transport simulations of processes in clays by using the Nernst—Planck equation: A benchmark revisited. Exact solution of the unidimensional Poisson—Boltzmann equation for a 1: 2 2: 1 electrolyte. Multicomponent diffusion modeling in clay systems with application to the diffusion of tritium, iodide, and sodium in Opalinus clay. Obtaining the porewater composition of a clay rock by modeling the in- and out-diffusion of anions and cations from an in-situ experiment.

Theory of the chemical properties of soil colloidal systems at equilibrium. Studying the migration behaviour of selenate in Boom Clay by electromigration.

### 1 Introduction

Sorption kinetics of strontium in porous hydrous ferric oxide aggregates: I The Donnan diffusion model. A general framework for ion equilibrium calculations in compacted bentonite. Ion equilibrium between montmorillonite interlayer space and an external solution-Consequences for diffusional transport. Solution of the Poisson—Boltzmann equation for surface excesses of ions in the diffuse layer at the oxide electrolyte interface. Sealing shales versus brittle shales: A sharp threshold in the material properties and energy, technology uses of fine-grained sedimentary rocks.

Molecular dynamics simulations of the electrical double layer on smectite surfaces contacting concentrated mixed electrolyte NaCl—CaCl 2 solutions. Molecular dynamics simulations of water structure and diffusion in silica nanopores. Anion exclusion and coupling effects in nonsteady transport through unsaturated soils: I Theory. Acid water—rock—cement interaction and multicomponent reactive transport modeling. General solution for Poisson—Boltzmann equation in semiinfinite planar symmetry. Modeling the long-term stability of multi-barrier systems for nuclear waste disposal in geological clay formations.

Structure of international simple glass and properties of passivating layer formed in circumneutral pH conditions. Molecular dynamics simulations of water structure and diffusion in a 1 nm diameter silica nanopore as a function of surface charge and alkali metal counterion identity. Microstructural investigation of gas shales in two and three dimensions using nanometer-scale resolution imaging. Diffusion of organic anions in clay-rich media: Retardation and effect of anion exclusion.

Non-equilibrium Thermodynamics. An interdisciplinary approach. Applying the squeezing technique to highly consolidated clayrocks for pore water characterisation: Lessons learned from experiments at the Mont Terri Rock Laboratory. A direct method for determining chloride diffusion coefficient by using migration test. Shortcomings of geometrical approach in multi-species modelling of chloride migration in cement-based materials. Physical modeling of the electrical double layer effects on multispecies ions transport in cement-based materials.

Effects of a thermal perturbation on mineralogy and pore water composition in a clay-rock: an experimental and modeling study. In situ chemical osmosis experiment in the Boom Clay at the Mol underground research laboratory. ANDRA underground research laboratory: Interpretation of the mineralogical and geochemical data acquired in the Callovian—Oxfordian Formation by investigative drilling.

Introducing interacting diffuse layers in TLM calculations: A reappraisal of the influence of the pore size on the swelling pressure and the osmotic efficiency of compacted bentonites. Chapter 8—Semipermeable membrane properties and chemomechanical coupling in clay barriers. Structure of nanocrystalline calcium silicate hydrates: insights from X-ray diffraction, synchrotron X-ray absorption and nuclear magnetic resonance. Quantitative X-ray pair distribution function analysis of nanocrystalline calcium silicate hydrates: a contribution to the understanding of cement chemistry.

Performance-based indicators for controlling geosynthetic clay liners in landfill applications. Ultrafiltration by a compacted clay membrane—II Sodium ion exclusion at various ionic strengths. Molecular dynamics simulations of water and sodium diffusion in smectite interlayer nanopores as a function of pore size and temperature.

Salt exclusion in charged porous media: a coarse-graining strategy in the case of montmorillonite clays. Diffusion of ionic tracers in the Callovo—Oxfordian clay-rock using the Donnan equilibrium model and the formation factor. Water and ion movement in thin films as influenced by the electrostatic charge and diffuse layer of cations associated with clay mineral surfaces.

Movement of water as effected by free energy and pressure gradients: II Experimental analysis of porous systems in which free energy and pressure gradients act in opposite directions. Movement of water as effected by free energy and pressure gradients: I Application of classic equations for viscous and diffusive movements to the liquid phase in finely porous media. Sorry, this product is currently out of stock. Flexible - Read on multiple operating systems and devices. Easily read eBooks on smart phones, computers, or any eBook readers, including Kindle. When you read an eBook on VitalSource Bookshelf, enjoy such features as: Access online or offline, on mobile or desktop devices Bookmarks, highlights and notes sync across all your devices Smart study tools such as note sharing and subscription, review mode, and Microsoft OneNote integration Search and navigate content across your entire Bookshelf library Interactive notebook and read-aloud functionality Look up additional information online by highlighting a word or phrase.

Institutional Subscription. Free Shipping Free global shipping No minimum order. Cites and analyzes mass transport equations developed for different membrane processes. Defines the mass transfer rate for first- and zero-order reactions and analytical approaches are given for other-order reactions in closed mathematical forms. Analyzes the simultaneous convective and diffusive transports with same or different directions. Powered by. You are connected as.

Connect with:. Use your name:. Thank you for posting a review! We value your input. Share your review so everyone else can enjoy it too. Your review was sent successfully and is now waiting for our team to publish it. Reviews 0. Updating Results. Mostly the Langmuir adsorption isotherms and the Maxwell Stefan diffusion model are applied to describe the transport through an aluminosilicate membrane Kapteijn et al.

The diffusion in ceramic mostly zeolite membrane takes place in narrow chan- nels both in the bulk channel phase and on the channel surface. The general form of the generalized Maxwell Stefan equation applied to surface diffusion is given as Wesselingh and Krishna, ; Krishna et al. The gradient of the thermodynamic potential can be expressed in terms of ther- modynamic factors as it was shown at beginning of this section Krishna, ; Kapteijn et al.

The combination of Eqs 1. The choice of the adsorption model determines the mathematical form of the thermodynamic factor obtaining according to sense from Eq. The sorption characteristics of the different sites can differ from each other Krishna et al.

A single-component adsorption also can be described by dual-site Langmuir model Zhu et al. For a binary mixture components 1 and 2 , the dual-site Langmuir isotherm can be given as Krishna et al. Equation 1. Note that Eq. To quantify the loading dependence, the model developed by Reed and Ehrlich is the most- often applied one.

It is widely accepted Thomas et al. This model takes into account molecular and Knudsen diffusion, as well as the contributions of viscous flux. One of the most advanced concepts is based on the assumption of activated configuration diffusion in the small pores [see Eq. Accordingly, the molar flux densities Ji for all components are as Thomas et al. Molecular diffusion 2. Knudsen diffusion 3.

Viscous flux For a single gas permeation through a membrane, the following flux can be obtained from Eq. The model is based on the idea that the chain elements arrange themselves randomly on a three-dimensional structure. The resulting equation for the activity of the solvent is a simple proportional function of the volume fraction of the solvent. The activity of a component in the membrane can be described according to Flory Huggins thermodynamics Flory, ; Lue et al. Its value can be positive or negative. After differentiating Eq.

Adapting Eq. In reality, according to Eq. The detailed discussion of these models is not a topic of this material, only the UNIQUAC approach, originally proposed by Abrams and Prausnitz , will be shown briefly. This model accounts for the different sizes and shapes of the molecules, as well as for the different intermolecular interactions between the mixture components, including polymeric components. References Abrams, D. Amundsen, N.

AIChE J. Baker, R. John Wiley and Sons, Chichester. Bird, R. John Wiley and Sons, New York. Bitter, J. Shell-Laboratorium, Amsterdam. Flory, P. Cornell University Press, New York. Gardner, T. Desalination , Geankoplis, C. Prentice Hall, New Jersey. Geraldes, V. Heintz, A. Concentration polarization, coupled diffusion and the influence of the porous layer. Hoang, D. Design 82, Kapteijn, F. Application of the generalized Maxwell Stefan equations. Krishna, R.

Lawson, K. Lonsdale, H. Li, S. C , Lue, S. Martinek, J. Mason, E. Elsevier, Amsterdam. Meuleman, E. Mulder M. Mulder, M. Calculations of concentration profiles. Nagy, E. Paul, D.

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## Basic Equations of Mass Transport Through a Membrane Layer - 2nd Edition

Methods 5, Reed, D. Schaetzel, P. Seidel-Morgenstern, A. Wiley-VCH, Weinheim. Silva, P. John Wiley and Sons, New Jersey, pp. Skoulidas, A. A study linking MD simulation with the Maxwell Stefan formulation. Langmuir 19, Thomas, S. Catalysis Today 67, Tuchlenski, A. Van de Graaf, J. Vignes, A. Wesselingh, J. Ellis Horwood, Chichester, U. Delft University Press, Delft. Wijmans, J. Zhu, W. Light alkanes in silicalite Molecules move at high speeds but travel extremely short distances before colliding with other molecules. The migration of individual molecules, therefore, is slow except at quite low molecular densities.

Because the molecules travel in random paths, mainly due to the frequent collision with other moving molecules, molecular diffusion is often called a random-walk process Geankoplis, But the real force can be obtained by means of thermodynamic consideration. If energy were dissipated in moving a molecule down the chemical potential gradient, the driving force, Fi, per molecule of species, i, would be Bungay et al.

In the simplest form of this theory, molecules are hard spheres. In a simplified treatment, it is assumed that there are no attractive or repulsive forces between molecules. A more accurate and rigorous treatment must consider the intermolecular forces, attraction and repulsion between molecules as well as different sizes of molecules A and B. Chapman and Enskog Hirschfelder et al. For a pair of nonpolar molecules, a reasonable approximation to the forces is the Lennard Jones function.

The effect of concentration of components is not included in Eq. Equation 2. Hence, the semiempirical method of Fuller et al. This method P can be used for mixtures of nonpolar gases or for a polar nonpolar mixture. But, as the solute becomes smaller, the approximation of the solvent as a continuous fluid becomes less valid. Geankoplis, , p. Each ion diffuses at a different rate. If the solution is to remain electrically neutral at each point, the cations and anions diffuse effectively as one component, and the ions have the same net flux.

The well-known Nernst Haskell equation for a dilute, single-salt solution can be used to predict the overall diffusivity DAB of the salt A in the solvent B Geankoplis, , p. This latter process is discussed in the next section. If the diffusion coefficient is independent of penetrant concentration, it can be determined by time lag or by equilibrium sorption measurements Vieth et al. In the case of time lag measurements, flux through a membrane is measured as a function of time when pressure is applied to one side of the membrane and vacuum is pulled at the other.

Equations for these cases are discussed in detail in Chapters 3 and In the two- or multicomponent separation, the coupling of the diffusion of the components occurs in most cases. The Maxwell Stefan approach for coupling is discussed in Chapter 3 in cases of pervaporation and gas separation by application of zeolite membrane Nagy, Another often recommended mass transport theory is the so-called Flory Huggins approach Meuleman et al.

This theory is especially applicable for organo- philic pervaporation of organic compounds in water. For polymers with high molecular weight, the activity in a membrane, according to the Flory Huggins approach, in a case of single-component transport, can be given as [see Eq. For detailed analysis of the mass transport for unary and binary systems, see Chapter 1. Basically, the following transfer rate can be defined depending on the pore size Soni et al. The flux is directly proportional to the difference of partial pressures.

Look at briefly the situation when the whole fluid is moving in bulk or convective velocity. Thus, for the mass transport through a membrane layer [see also Eqs 1. Bird, B. Bungay, P. Reidel Publishing Company, Vol Barrer, R. Faraday Soc. Crank, J. Academic Press, London. Dechadilok, P.

## Basic Equations of the Mass Transport through a Membrane Layer

Deen, W. Fuller, E. Geankoplis, Ch. Prentice Hall, Upper Saddle River. Hirschfelder, J. Mavrovouniotis, G. Colloid Interface Sci. Qin, Y. Smart, J. Part II. Experimental results. Soni, V. Computers Chem. Vieth, W. For a description of this process, the solution-diffusion model is applied.

That means that the transported component is adsorbing on the solid interface and then it is transported by molecular diffusion through the membrane layer to its external surface, and from here through an external mass transfer resistance to the bulk permeate phase. Note that the concentrations are in equilibrium at the interface, between the continuous phase and membrane layer, and that this sorption process is an instantaneous process.

Most of the membrane modules used for separation are capillary membrane hollow-fiber modules. The question arises, under which conditions the mass transport through a cylindrical membrane layer can be considered as a plane layer. The diffusional mass transfer equation through a cylindrical membrane is defined in Chapter 6. The selective or skin layer of an asymmetric membrane is very thin; its thickness falls generally in the range of 0. Besides these special cases, the mass transport through a membrane layer through plate-and-frame, tubular, and spiral-wound modules should be regarded as that through a plane interface.

For a longer time, a steady state is reached in which the concentration in the membrane remains constant. After integration of Eq. Because different investigators can use different units, the permeability constant can have different units and even different definitions. Look at the situation when feed phase and also sweep fluid permeate side; this is the case during dialysis or often during pervaporation are flowing on the two sides of the membrane layer. This situation is illustrated by Figure 3. The general solution of Eq. When membranes are used for pervaporation dehydration, or organic organic separation, appreciable mem- brane swelling usually occurs, and both the partition and diffusion coefficient become concentration dependent.

Therefore, the classic solution-diffusion theory should be modified to adapt to the generally swollen pervaporation membranes. When a membrane is swollen or plasticized by transporting species, the interac- tion between polymer chains tends to be diminished, and the membrane matrix will therefore experience an increase in free volume.

It is generally true that in a given membrane, increased free volumes correspond to increased diffusion coefficients of the penetrants. When a membrane is plasticized by more than one species, the diffusion coefficient of a species is facilitated by all the plasticizants. The plasticization coefficient of the less-permeable species can be neglected during dehydration processes, because dehydration membranes generally show overwhelming affinity for water, and the concentration of the less-permeable species in the membrane is negligibly small.

The diffusion coefficients of both the species in the membrane are thus dependent on the concentration of water in the membrane phase alone. If the D is concentration dependent, Eq. It is important to note that the concentration depends on the concentration dependency of D, thus its value is not known, and thus, the real concentration distribution can be calculated by iter- ation, supposing a concentration distribution at beginning.

Starting from this value, we should then calculate the value of Dm and then the concentration distribution. About three to four calculation steps are enough to get the correct concentration distribution by means of the real diffusion coefficients. Knowing the value of J, the S value can be calculated in order to predict the concentration distribution by one of the boundary conditions, for example replacing the limiting case of Eq. The average value of the diffusion coefficient can be calculated by Eq.

## PEM Fuel Cells, Modeling

It can be seen that the relatively large value of the external mass transfer resistance can strongly diminish the effect of the exponentially increasing diffusion coefficient. Knowing the values of S and J, the concentration distribution can be calculated by means of Eq. Thus, a general solution of the prob- lem will be shown in this section. The solution methodology will be discussed in this chapter.

See e. The sorption coefficient can also vary as a function of the concentration, but this dependency is much lower than that of the diffusion coefficient Chandak et al. The main types of this dependency are discussed briefly here, namely linear and the Langmuir-type absorption isotherms. Alternatively, a more thermodynamically rigorous model like the extended Flory Huggins model can also be used. The role of the concentration dependency is especially important when the external mass transfer resistance cannot be neglected. The other possibility is that by rearranging this equation, one can get a third-order algebraic equation that has an analytical solution.

It has been demonstrated that the sorption isotherms of small molecule penetrant gases such as carbon dioxide, methane, argon, etc. The dual mode sorption model is widely used to describe such behavior. The following equation can be obtained from Eq. Differentiating Eq. Applying Eq.

To develop Deff as a function of position in the membrane layer, after integra- tion of Eq. By means of Eq. The diffusion coefficient is also different, but its value is constant inside of every sublayer. A strong coupling of diffusion of components to be separated takes place in the Maxwell Stefan approach Wesselingh and Krishna, This approach could be applied to describe mass transport, with strongly concentration- dependent diffusion, during pervaporation of binary water alcohol mixture, with low carbon number Heintz and Stephan, , as well as during membrane extraction of organic components Kubaczka et al.

Separation by means of zeolite, or, generally, of inorganic membranes is another important group of membrane separation processes where the Maxwell Stefan approach to mass transfer is recommended Krishna and Wesselingh, ; van den Graaf et al. This theory is especially applicable for organophilic per- vaporation of organic compounds in water.

For the estimation of the countersorp- tion diffusivity, Krishna and Wesselingh and Bitter proposed to use a generalized form of Vignes equation, which also means strong concentration- dependent diffusion coefficients. The differential mass balance equations, for binary, diffusional mass trans- port processes, can generally be given by Eqs 3. The solution of these equations can only be done by numerical methods, which makes their applications difficult.

Recently, Nagy gave an analytical approach for the solution, but the equations developed are very complex. The aim of this work was to develop a much simpler mathematical model that defines the concentration distribution of components in the cases of single component as well as that of binary, coupled diffusion processes in the membrane layer and the mass transfer rates of the diffusing components at the membrane interface, by means of explicit mathematical equations.

This approach should generally be used independently of what function the diffusion coefficient of components could depend on the concentrations. As a practical example, the binary mass transport through zeolite membrane will be shown by applying the well-known Maxwell Stefan approach. Obviously, the model can be used for multilayered membranes having space-dependent mass transport parameters.

Diffusion Through a Plane Membrane Layer 63 3. Basically, there are two main theories for the coupling diffusion process, namely the Maxwell Stefan theory Heintz and Stephan, ; van den Graaf et al. It also should be men- tioned that the Vignes and the modified Vignes equations can be used Bitter, for the concentration dependency of the real diffusion coefficients given by Eq. All theories mentioned above involve the above effective diffusion coefficients, and the component transport can be described by the differential equations 3.

The schematic diagram of the physical model applied in order to get an analytical approach of the solution is illustrated in Figure 3. Both cases involve large parts of membrane processes depending on the hydrodynamic conditions of the streaming phases, as well as the transfer rate inside of the membrane. During gas mixture separation, it could often be neglected, while during liquid separation process often not.

In this text, the mass transfer rate without external mass transfer resistance will be shown. This case can be used for mass transport of gas components through a membrane. Their solution is well known, and they are given in Eqs 3. Assuming N sublayers in the membrane, one can get N algebraic equations for the concentration distribution of both components. Each component has 2N parameters to be determined Nagy, These parameters can be calculated by means of the 4N boundary conditions defined by Eqs 3.

The algebraic equation system containing 2N equations, obtained using Eqs 3.

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Both the concentrations and mass transfer rates were calculated by means of the predicted value of DAB, using the modified Vignes equation given by Eq. The data used for calcula- tion are given in Table 3. The overall pressure was kept to be 1 kPa in this side. Both the concentration and mass transfer rates Figure 3. As can be seen, the mass transfer rate of methane is strongly affected by the friction between the molecules Figure 3. The concentrations of the components alter drastically the diffusion rate of the other components.

The effect of the adsorbate adsorbate interaction is also essential. Note that here DAn and DBn diffusion coefficients are not average values, their values should be fitted to the real concentration in the membrane. Thus, the calcula- tion of the concentration distribution or the mass transfer rate needs a few iteration steps in order to obtain the real diffusion coefficients to the real concentration values. Equations 3. The diffusion coefficient of water and ethanol in the mem- brane used for our calculations, as a function of concentrations, were measured by Hauser et al. The value of D12 was also predicted by Heintz and Stephan The interface concentrations of the upstream side at y 5 0 were also used measured values.

The external mass transfer resistance and that for the porous support layer were neglected. As seen in Figure 3. The effective different diffusion coefficients, according to Eqs 3.

### Review ARTICLE

To calculate the amount of species transferred into the membrane, it is first necessary to determine the concentration distribution of the transferred species within the membrane layer as a function of position and time. This transitional state is described by Eq. This is possible only if both sides of Eq. Equation 3. The boundary conditions given at Y51 in Eq.

The use of the boundary condition defined at Y51 in Eq. Therefore, when n 5 m, the summation drops out and Eq. Diffusion Through a Plane Membrane Layer 77 3. At small values of time, the component does not penetrate very far into the membrane layer. Under these circumstances, it is possible to consider the slab as a semiinfinite medium in y-direction.

Multiplication of Eq. In certain cases, the Boltzmann transformation may be employed to convert this to an ordinary differential equation as it is done in Eqs 3. The solution is discussed by Follain et al. For a numerical solution, see Section A. References Baker, R. Wiley, Chichester. Shell- Laboratorium, Amsterdam. Chandak, M. Carslaw, H. Clarendon Press, Oxford.

Favre, F. Follain, N. Ghoreyshi, A. Hauser, J. Fluid Phase Equilib. Koros, W. Kubaczka, A. Shah, M. Slattery, J. Cambridge University Press, Cambridge. Elsevier, New York. Hanser Publisher, Munich. Delft University Press, The Netherlands. A number of reactions have been investigated by means of this process, such as dehydrogenation of alkanes to alkenes, partial oxidation reactions using inorganic or organic perox- ides, as well as partial hydrogenations and hydration. As catalytic membrane reactors for these reactions, intrinsically catalytic membranes can be used e.

In the majority of the above experiments, the reactants are separated from each other by the catalytic membrane layer. In this case, the reactants are absorbed into the catalytic membrane matrix and then transported by diffusion and in special cases by convection from the membrane interface into catalyst particles where they react. Mass transport limitation can be experienced with this method, which can also reduce selectivity.

The application of a sweep gas on the permeate side dilutes the permeating component, thus increasing the chemical reaction gradient and the driving force for permeation Westermann and Melin, For their description, two types of membrane reactors should generally be distinguished, namely intrinsically catalytic membranes and membrane layers with dispersed catalyst particle, either nanometer-sized or micrometer-sized catalyst particles.

Both approaches, namely the heterogeneous model for larger catalyst particles and the homogeneous one for submicron particles, will be applied for mass transfer through a catalytic mem- brane layer. The boundary conditions can depend on the external mass transfer resistance as discussed here. As mentioned, the catalytic membrane can be intrinsically catalytic or the membrane matrix can be made catalytic by dispersed catalytic particles.

For a membrane with dispersed catalyst particles, the Q source term should involve the mass transport in the membrane matrix to the catalyst particle and the simultaneous internal transport, as well as the internal chemical reaction. Accordingly, the source term can be strongly different for a membrane reactor with dispersed catalyst particles or that for an intrinsically catalytic membrane layer. The mathematical description of the mass transport through these membrane layers can be different depending on the size of the catalytic particle. Thus, presentation of the mass transport equations is divided into two parts, namely: 1.

Mass transport through intrinsically catalytic or nanometer-sized catalytic particles are dispersed in the membrane layer; in this case, it can be assumed that the mass transport inside the catalytic particles or the mass transport to the catalytic interface is instanta- neous and catalytic particles can be located in every differential volume element of the membrane; accordingly, the membrane can be regarded as a continuous catalytic layer.

The dispersed catalytic particles fall into the micrometer-sized regime, the internal mass transport mechanism, inside of catalyst particles, must be taken into account. In this case, the so-called heterogeneous model should be used, which takes into account the internal mass transport as well. It is assumed that catalyst particles are placed in every differential volume element of the membrane reactor. The reactant first enters in the membrane layer and from that, it enters into the catalyst particles where the reaction of particles is porous as active carbon, zeolite Vital et al.

Consequently, the mass transfer rate into the catalyst particles has to be defined first. In this case, the whole amount of the reactant transported in or on the catalyst particle will be reacted. Then this term should be placed into the mass balance equation of the catalytic membrane layer as a source term.

Thus, the differential mass balance equation for intrinsically catalytic membranes and membranes with dispersed nanometer-sized particles differ only by their source term. The cylindrical effect can be significant only when the thickness of a capillary membrane can be compared to the internal radius of the capillary tube as shown by Nagy On the other hand, the application of a cylindrical coordinate hinders the analytical solution for first- or zero-order reactions as well.

Thus, the basic equations will be shown here for plane interface and in the section 6. Catalyst with dispersed particles; reaction takes place inside of the porous particles: The differential mass balance equation for the catalytic membrane can be given as! It is assumed in Eq.

The reactant diffuses in the catalyst particle and it reacts. This case, when the reaction occurs at the catalyst surface, is also briefly discussed later. The membrane layer with nanometer-sized catalyst particles is illustrated in Figure 4. For a catalytic membrane with dispersed nanometer-sized particles, the mass transfer rate into the spherical catalyst particle must be defined. According to Figure 4. Thus, one can obtain Nagy et al. From Eqs 4. Reaction occurs on the interface of the catalytic particles Nagy, : It often might occur that the chemical reaction takes place on the interface of the particles, for example, in cases of metallic clusters; the diffusion inside the dense particles is negligible.

The above model is obviously a simplified one. Figure 4. The k1 value was calculated by Eq. Both the reaction modulus and the catalyst holdup can strongly affect the inlet mass transfer rate of the membrane layer. At the end of this subsection, the limiting cases will also be given briefly. Note that the overall mass transfer rate can also be obtained by means of resistance-in-series model. The mass transfer rate can similarly be obtained for the case when the outlet concentration.

For this, the operating conditions should be chosen correctly. The concentration distribution is illustrated in Figure 4. The solution of the mass balance equation is as given in Eq. Figures 4. That is why it will be briefly discussed here for the sake of completeness. In the case of dispersed fine catalyst particles, the specific mass transfer rate should be given in order to use it as a reaction term in the mass balance equation given for the membrane phase. For the solution of Eq.

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The physical model to get an analytical approach is illustrated in Figure 4. Thus, one can get a second-order differential equation with a linear source term that can be solved analytically. The N algebraic equations obtained can be solved using the well-known Cramer rules. Note that in order to deter- mine the T1 and S1 parameters, the average concentration of the B component should be known. For this, its correct value should be used. Thus, for prediction of the J value, the concentration of component B must be known. It is easy to learn that a trial-and-error method should be used to get the component concentra- tions alternately.

Three to four calculation steps are enough to get the correct value of one of the components. The concentration of a component gradually and automatically approaches its correct value belonging to a given concentration of the other component. Steps of calculation of concentration of both components can be as follows: 1. A starting concentration distribution for component B should be given and one calculates the concentration distribution of component A applying Eqs. The indices of sublayers for component A have to be changed and adjusted to that of B starting from the permeate side of membrane, i.

These three steps should be repeated three to four times until concentrations of both com- ponents do not change anymore. Knowing the T1 and S1, the other parameters, namely Ti and Si i 5 2, 3,. From that, its mass transfer rate can easily be calculated by a similar manner as for the component A. The reaction modulus was chosen five times higher for component B, thus its value of component A was relatively low; it varied between 0.

As can be seen, the concentration of component A varies only in lesser extent. Accordingly, its concentration gradient will be zero at Y51 model B. Therefore, it seems important to analyze this mass transport process. The starting equations are the same as in the previous case.

The concentration distribution is plotted in Figure 4. Strong effects of the reaction rate can be observed for component B.