Geometrical analysis of dynamic problem on the membrane transport using spectral solution

The problem analyzed in this paper is a specific application of the composite membrane. General diffusion and convection formulation is presented for the dynamic problem. The spectral analysis considers convective transport of a single solute species across a one dimensional membrane system. The solution is obtained using operator theoretic methods. The geometrical structure of the spectrum of the operator is determined for the complete range of the various parameters including the distribution coefficient, the convective velocity and the diffusion coefficient. The structure of the spectrum allows a complete characterization of all the eigenvalues of the system in terms of all of these physical parameters. Calculation of the first eigenvalue for a number of cases shows its variation with the convective velocity for various medium porosities and allows a priori estimates of the profiles.     

Transport of swelling penetrants in glassy polymers: Influence of convection

The transport of a solute by diffusion into a glassy polymer can lead to swelling of the material. For certain types of polymers, a sharp interface is formed between the swollen region and the glassy core. When the density of the swollen material is much smaller than the density of the glass, a significant convective mass-average velocity is generated within the sample. Previous models have neglected the role this convection plays in solute transport and in the proper calculation of the sample dimensions as a function of time. In this paper, we study the contribution of convective terms to the solute transport process, including the motion of the swollen polymer/solution interface. We also compute the eulerian strains that result from the calculated velocity fields and the stresses that would be generated if a linear viscoelastic model is used as a constitutive equation relating the stress to the strain. We show that serious errors can be generated in the calculations if convective terms are neglected. Furthermore, a comparison of the strains and stresses acting on the polymer with those acting on the mixture of solute and polymer shows that they can be significantly different. The stresses and strains acting on the polymer alone offer the most rational physical picture of the material deformation.     

Characterization of transport across cellulose acetate membranes in the presence of strong solute–membrane interactions

Certain organic solutes, including phenol, undergo anomalous enrichment when hyperfiltered through cellulose acetate membranes: the solute concentration is higher in the permeate than in the feed solution. A number of existing theoretical approaches describing hyperfiltration phenomena are presented and their merits and limitations upon application to the transport of phenol discussed. A new two-parameter transport relationship is derived based on an extension of the solution–diffusion model. The enrichment, or negative solute rejection by the membrane, is predicted to occur whenever the pressure-induced solute permeation velocity exceeds that of water. By acknowledging and incorporating the effect of pressure on the chemical potential of the solute, the present extended solution–diffusion model relationship successfully describes hyperfiltration data of phenol in homogeneous and asymmetric cellulose acetate membranes provided the contribution of convective flow to the overall solute transport is insignificant. In addition to the transport parameters of the extended solution–diffusion model, the transport parameters of the phenomenological, Kedem–Spiegler, and combined viscous flow–frictional relationship are evaluated from hyperfiltration data obtained with 0.05 and 0.1 wt % phenol feed solutions and homogeneous cellulose acetate membranes of different acetyl content.     

Rapid Salt Exchange by Coupled Ultrafiltration and Dialysis in Anisotropic Hollow Fibers

Abstract

Anisotropic hollow fibers allow construction of a dialyzing system that provides extremely large membrane surface in a small laboratory-sized system. Possessing the added property of high ultrafiltration flux, these fibers reduce salt exchange times from days to hours. In this system the exchange of salt by dialytic transport is largely unaffected by recirculation rate, solute type, or content, but is strongly affected by those variables which affect molecular diffusion, such as microsolute size and temperature. In contrast, diafiltration (convective salt removal by ultrafiltration), which primarily relates to solvent transport through the membrane, can be changed by operating pressure, polarizability of the macrosolute, as well as those conditions which tend to influence this latter phenomenon.     

Modeling of the variations of permeate flux,concentration polarization,and solute rejection in nanofiltration system

A numerical model is presented and experimentally validated for predicting the local concentration polarization and the related separation performance of nanofiltration (NF) system. The model combines computational fluid dynamics for describing the transport phenomena in NF channel, with Spiegler-Kedem-Katchalsky model for considering the permeation properties through NF membrane. Particular attention is given to the modeling of spatially varied solute rejection and solute transport through membrane, representing essential distinctions from the modeling of reverse osmosis (RO) membrane. Also, an experimental-numerical framework is proposed to determine model parameters related to solute transport, including reflection coefficient and solute permeability as functions of feed solute concentration. The usefulness of this model is highlighted by predicting concentration polarization under different conditions related to operations, membrane systems (NF vs. RO), and solute types (NaCl vs. MgSO₄). Also, the contributions by convection and diffusion to solute transport are clarified, benefiting by the modeling of solute transport. © 2018 American Institute of Chemical Engineers AIChE J, 65: 1076–1087, 2019     

REVIEW OF REVERSE OSMOSIS MEMBRANES AND TRANSPORT MODELS

After a brief introduction to membrane processes in general, and the reverse osmosis process in particular, the structure and properties of membranes and membrane transport theory are described. The mechanism of salt rejection and transport properties of membranes are discussed in detail. Solubility, diffusivity, and permeability of membranes to solutes and solvents are reviewed critically and compared with each other. Special attention is given to two particular types of membranes, cellulose acetate (CA) and aromatic polyamide (AP) membranes, which are often used for water desalination.

The major portion of this article is devoted to the review and discussion of membrane transport theory with application to the reverse osmosis and ultrafiltralion processes. It is shown that the solvent flux can be represented reasonably well by linear models such as the solution-diffusion model (Lonsdale, et al., 1965). The contribution of pore flow to the solvent flux is small. The solute flux, however, is not linearly dependent on the driving forces and one has to solve the differential equation of transport within the membrane which results in models such as the Spiegler-Kedem (1966) or the finely-porous (Merten, 1966) models. When the wall Peclet number is small, Pe_w =uτδ/D_sw ?1, (D_sw = bD_e one can linearize the nonlinear models. This requirement is not satisfied in most practical cases. Furthermore, the pore flow has significant effect on the solute flux equation and thus it can not be neglected.

The ambiguities that exist in the literature concerning the types of fluxes are discussed. The fluxes used in models derived from irreversible thermodynamics are purely diffusive (concentration and pressure diffusion) and they do not contain any convective effects; whereas the experimentally observed fluxes are the total fluxes with respect to the membrane which consist of a diffusive flux and a convective flux. A new model, based on irreversible thermodynamics, is derived which includes a convective term.

A membrane model is especially useful when the transport coefficients which define the model are not functions of the driving forces, i.e., pressure and concentration gradients. The coefficients in the solution diffusion and sotution-diffusion-imperfection (Sherwood, et al., 1967) models are functions of both pressure and concentration, while the coefficients in the Kedem-Katchalsky (1958) model are relatively insensitive to pressure and concentration. The nonlinear model of Spiegler-Kedem (1966) further improves the Kedem-Katchalsky model.     

Solute–membrane interactions in hyperfiltration

The performances of several composite membranes (PEC-1000, Teijin, HR-95, HR-98) and one asymmetric membrane (Solrox SC-200) in hyperfiltration are compared at 25°C using different aqueous feed solutions (0.1M and 0.5M NaCl, 0.5M 1,3-and 1,4-dioxan, 0.1M benzyl alcohol and 2-methoxybenzyl alcohol, 1,2- and 1,4-butandiol, and Triton feed solution). The effects of solute dissociation, polarization, and hydrogen bonding ability on solute permeability are discussed; steric effects are also being considered. Strong solute–membrane interactions are exhibited in hyperfiltration by systems with distinct hydrogen bonding capabilities of the solute with functional sites of the membrane matrix. Knowledge of solute–membrane interactions can be useful for elaborating separation and transport mechanisms.     

An analytical relationship of concentration‐dependent interfacial solute distribution coefficient for aqueous layer freeze concentration

Freeze concentration (FC) is a subzero temperature solute concentration procedure, favoring the retention of high quality compounds such as food ingredients and biological materials. It is known that modeling solute inclusion in the ice layers or ice crystals formed in a convective environment requires the solute distribution coefficient function. The fluid flow velocity, ice‐growth rate and solute concentration are influential on this function. Some literature has reported certain expressions of the function, which are relatively complex. Here, an explicit format of this function has been derived for single solute system, and found to be satisfactory in correlating a wide range of experimental data on sucrose solutions for both the controlled flow layer crystallization process (flow in between two cooling plates) as well as the falling film crystallization process. This expression has captured the fundamental aspects of mass transfer, and it is relatively simple which should be very useful for correlating FC parameters and for simulating the layer FC processes. © 2015 American Institute of Chemical Engineers AIChE J, 61: 1334–1344, 2015     

TAYLOR DISPERSION OF MULTICOMPONENT SOLUTES

Coupled transport of multicomponent solutes in globally continuous systems is considered in the framework of the Generalized Taylor dispersion theory. Coupling between transports of n different species at the local (or micro-) scale, is considered to result from first-order irreversible surface reactions occurring on the local space boundaries, or from the off-diagonal terms of the solute diffusivity matrices.

General expressions are obtained for the global effective (long-time) solute dispersion matrix cofficients: mean global scalar reactivity, velocity vector and dispersivity dyadic.

The effect of surface chemical reactions is to partition the matter between different solute constituents. This is manifested in a coupling of the global transport coefficients, which may be mathematically removed by a linear (canonic) transformation applied to the effective global transport equation. This type of coupling does not exist for inert solutes.

The second type of the global coupling is represented by the off-diagonal terms of the global velocity and dispersivity matrices. It exists for both reactive and inert solutes. This coupling stems from the convective dispersion process (dependence or the global velocity vector on the local space coordinate). Is shown to be irremovable from the global transport equation by any linear transformation via the solute partition matrix. In the canonic form of the global equation the irremovable coupling is manifested by the traceless parts of the global solute velocity matrix and the global solute dispersivity.

The solution scheme is illustrated by calculating the mean global diffusivity of a solute consisting of two components, transport of which is coupled at the microscale via the molecular diffusivity matrix. At the macroscale the coupling is shown to be represented by negative off-diagonal terms of the global diffusivity matrix,     

Hydrodynamic dispersion of reactive solute in a Hagen-Poiseuille flow of a layered liquid

An analysis of the solute dispersion in the liquid flowing through a pipe by means of Aris-Barton's ⒈method of moments',under the joint effect of some finite yield stress and irreversible absorption into the wall is presented in this paper.The liquid is considered as a three-layer liquid where the center region is Casson liquid surrounded by Newtonian liquid layer.A significant change from previous modelling exercises in the study of hydrodynamic dispersion,different molecular diffusivity has been considered for the different region yet to be constant.For all time period,finite difference implicit scheme has been adopted to solve the integral moment equation arising from the unsteady convective diffusion equation.The purpose of the study is to find the dependency of solute transport coefficients on absorption parameter,yield stress,viscosity ratio,peripheral layer variation and in addition with various diffusivity coefficients in different liquid layers.This kind of study may be useful for understanding the dispersion process in the blood flow analysis.     

Piezodialysis

In this paper a previously developed model for a piezodialysis membrane is summarized and extended. Relationships between transport properties of resins and transport properties of a composite membrane are derived. These relationships provide considerable insight into the piezodialysis process and some useful direction in the search for better piezodialysis membranes. A relationship between geometric factors of the membrane and test apparatus and the effective circulating current path length is presented. Performance of a module which demonstrates continuous desalination by piezodialysis is described.     

Convective and diffusive mass transport through anisotropic,capillary membrane

Concentration and/or space-dependent diffusive and convective mass transport, through capillary, anisotropic membrane layer, dense or porous membrane, were investigated. A quasi-analytical solution is presented in order to estimate the concentration distribution in the membrane layer and the total mass transfer rates, namely the sum of the diffusive and convective ones. These properties have been given in closed, explicit, mathematical forms which make the calculation process very simple. The diffusion coefficient can vary with the concentration and/or space coordinate, while the convective velocity can vary with the space coordinate. It was shown that the change of the Pe-number can have strong effect on the mass transport process, thus its effect must not be neglected. The model developed can be applied by any mathematical function of the variation of the mass transport parameters.     

The effect of electrophoretic convection for the separation of solute in the chromatography column (I)

A theoretical model has been derived in an electrophoretic packed column where an electric potential is applied to a column in the axial direction. The effect of electrophoretic convection in gel particles packed in the column significantly contributes to the separation of large polyelectrolytes because the conformation of polyelectrolyte quickly orients in the field direction. The dependence of the transport in the gel particle upon field intensity and molecular size aids in understanding the transport of polyelectrolyte in the packed column, since the convective velocity of polyelectrolyte is accelerated inside a porous gel particle. There are few convection studies of large poly-electrolyte in a column packed with porous gel particles under an electric field for the separation. Convective-diffusive transport of a large polyelectrolyte is analyzed using Peclet number described by electrophoretic mobility and diffusion coefficient measured experimentally. The separation of two different polyelectrolytes in the packed column is performed using a value ofPe _f/Pe_g of individual polyelectrolyte by molecular size and an electric field. The purpose of this paper is to study the separation of solute from a mixture in the column using the physicochemical properties in the gel particle which are measured experimentally.     

Ultrafiltration on Sizing Agent Solution in Tubular-Membrane Module

Ultrafiltration of an aqueous solution of carboxymethyl cellulose (CMC) was carried out in a tubular-membrane model made of ZrO₂/carbon. Water was forced through the macroporous membrane as the permeate, while CMC was concentrated and recovered as the retentate. Correlation equations for calculating the permeate flux of membrane ultrafiltration were derived based on the resistance-in-series model. Correlation results were confirmed by the experimental data. Experimental results showed that the permeate flux increases as transmembrane pressure or fluid velocity increases, but decreases when feed concentration increases. Because membrane ultrafiltration is a pressure-driven process, high cross-flow velocity enhances the mass transfer coefficient of the solute and high solution concentration increases the thickness of the concentration polarization layer.     

膜萃取过程的传质特性研究

膜萃取是一种新型的分离技术。本文在中空纤维膜器中研究了膜萃取过程的传质特性。通过四种不同体系的实验,求取了基于水相的总传质系数,提出了求算膜萃取过程中各分传质系数k_(?)、k_(?)、k_(?)的半经验关联式。研究表明,减小膜阻可以强化膜萃取过程,提高过程的总传质系数.比较和分析膜萃取过程中各部分传质阻力,可以看出,对于萃取相平衡常数m1的体系应选用疏水膜器,对于m1体系则应选用亲水膜器。     

Control of Solids Uptake by Convective Drying Prior to Osmotic Processing of Foods

Solute impregnation constitutes a major drawback to the commercial use of osmotic dehydration of food and several methods to control solute uptake have been proposed. In this work the effect of a convective drying step before osmotic treatment in minimizing solute uptake by the material is analyzed. Results indicate that the initial convective step is a useful way to control solute uptake of apple without increasing total operation time. Reduction of solute uptake using an initial convective step of 30 min varied between 75 and 85% with respect to the samples submitted to a single osmotic treatment with the same duration. It was determined that an initial convective treatment of 30-60 min does not modify total operating time and provides an efficient control of solute gain by the material. This combined dehydration process may be an attractive alternative for producing a new generation of vegetable snacks.     

反渗透过程溶质脱除率方程

<正>溶解扩散理论、摩擦模型和表面力-孔流模型假设稳态时膜中溶质通量恒定“-”。本文分 析了该假设存在的理论依据,并基于膜的吸附-扩散模型’‘’建立了反渗透过程溶质脱除率方 程。     

Fundamentals of membrane transport

A review is presented to give a generalized membrane transport theory based on the principles of nonequilibrium thermodynamics. This theory is then used to develop specific flux equations for gas separation, pervaporation, osmosis, reverse osmosis, nanofiltration, ultrafiltration, microfiltration, dialysis, and electrodialysis. All membrane processes suffer from boundary layer mass transfer resistances caused by concentration polarization. The convective motions parallel and perpendicular to the membrane surface are distinguishable, and the former becomes more relevant than the latter in the boundary layer mass transfer. The modified P??clet number is introduced and its importance is discussed in characterizing the boundary layer mass transfers of various membrane processes. Many different transport mechanisms through membrane itself are reviewed including the solution-diffusion model, pore model, permeation through composite membranes, and transport through inorganic membranes. Finally, the differences between membrane mass transfer and other mass transfer are delineated, including a discussion of negative mass transfer coefficient.