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Theories of membrane transport and the application of non-equilibrium thermodynamics to transport processes have been described by Meares et al(9).

Many of the earlier treatments of membrane transport use the Nernst-Planck equations to describe the relationships between the flows of the permeating species and the forces acting on the system(10,63) according to Meares et al According to these equations the flux Jj of species i at any point is equal to the product of the local concentration cj of i, the absolute mobility uj of i, and the force acting on i. This force has been identified with the negative of the local gradient of the electrochemical potential 11; of i. Thus, at a distance x from a reference plane at right angles to the direction of unidimensional flow through a membrane
The electrochemical potential of i can be divided into its constituent parts giving in place of equation eq. ( where Yil Vil Zil p, and 1Jr represent the activity coefficient, the partial molar volume, the valence charge on i, the hydrostatic pressure, and the electrical potential, respectively. R is the gas constant, T the absolute temperature, and F the Faraday. It is apparent from eq. ( that the Nernst-Planck equations make use of the Nernst-Einstein relation between the absolute mobility uj and the diffusion coefficient OJof species i. This is On replacing the electrochemical mobility in eq. ( by the diffusion coefficient, the more usual form of the Nernst-Planck flux equation is obtained according to Meare .
On the basis of the Nernst-Planck equations, the flow of species i is regarded as unaffected by the presence of any other permeating species except in so far as the other species either influences the force acting on i by, for example, affecting the values of Yi or lJr,or alters the state of the membrane and hence alters the value of 01′ To obtain relationships between the flows of the permeating species and the observable macroscopic differences in concentration, electrical potential, and hydrostatic pressure between the solutions on the two sides of the membrane, it is necessary to integrate the Nernst-Planck equation (eq. for each mobile component across the membrane and the membrane/solution boundaries. In order to carry out this integration an additional assumption has to be made. The differences between the various treatments derived from the Nernst-Planck equations lie in the different assumptions used. For example, in the theory of Goldman (63), which is widely applied to biological membranes, it is assumed that the gradient of electrical potential dlJr/dx is constant throughout the membrane. It is usually assumed also that thermodynamic equilibrium holds across the membrane/solution interfaces and that the system is in a steady state so that the flows Jj are constant throughout the membrane. Generally these integrations do not lead to linear relationships between the flows and the macroscopic differences of electrochemical potential between the two bathing solutions.
The main disadvantage of the Nernst-Planck approach according to Meares(9)is that it fails to allow for interactions between the flows of different permeating species. Such interactions are most obvious when a substantial flow of solvent, usually water, occurs at the same time as a flow of solute. For example, during the passage of an electric current across a cation-exchange membrane, the permeating cations and anions both impart momentum to the water molecules with which they collide. Since the number of cations is greater than the number of anions, the momentum imparted to the water by the cations is normally greater than the momentum imparted by the anions and an electro-osmotic flow of water is set up in the direction of the cation current. The resultant bulk flow of the water has the effect of reducing the resistance to the flow of cations and increasing the resistance to the flow of anions. This flow of water occurs under the difference of electrical potential and in the absence of a concentration gradient of water. The appropriate Nernst-Planck equation would predict no flow of water under these conditions according to Meares et al Furthermore the flows of cations and anions differ from those which would be predicted from the respective Nernst-Plank equations on account of the effect of the water flow on the resistances to ionic flow.
This effect of solvent flow on the flows of solute molecules or ions can be allowed for by adding a correction term to the Nernst-Planck equations(9). Thus, it can be written where v is the velocity of the local centre of mass of all the species(ll). The term cjv is often called the convective contribution to the flow of i and some authors have preferred to define v as the velocity of the local centre of volume.
The addition of this convection term to the Nernst-Planck equation for the flow of a solute is probably a sufficient correction in most cases involving only the transport of solvent and nonelectrolyte solutes across a membrane in which the solvent is driven by osmotic or hydrostatic pressure according to Meares et al The situation is much more complex when electrolyte solutes are considered according to Meares et al Even at low concentrations the flows of cations and anions may interact strongly with each other. Interactions between the different ion flows may be of similar size to their interactions with the solvent flow. Under these circumstances the convection-corrected Nernst-Planck equations may still not give a good description of the experimental situation regarding the ion flows.
The theoretical difficulties arising from interacting flows can be formally overcome by the use of theories of transport based on nonequilibrium thermodynamics. Such theories are described in the next section.
Since the original papers of Staverman(12)and Kirkwood(64),•many papers have appeared on the application of nonequilibrium thermodynamics to transport across synthetic and biological membranes. In particular, major contributions have been made by Katchalsky, Kedem, and co-workers. In view of the appearance of extensive texts(1314),. this account is intended only as a brief summary of the general principles.
The theory of nonequilibrium thermodynamics allows that, in a system where a number of flows are occurring and a number of forces are operating, each flow may depend upon every force. Also, if the system is not too far from equilibrium, the relationships between the flows and forces are linear. Therefore, the flow Jj may be written as follows where the Xk are the various forces acting on the system and the Ljk are the phenomenological coefficients which do not depend on the sizes of the fluxes or forces. The flow Jj may be a flow of a chemical species, a volume flow, a flow of electric current, or a flow of heat. The forces Xk may be expressed in the form of local gradients or macroscopic differences across the membrane of the chemical potentials, electric potential, hydrostatic pressure, or temperature. If a discontinuous formulation is used so that the macroscopic differences in these quantities across the membrane are chosen as the forces, then the Lik coefficients in eq. ( are average values over the membrane interposed between a particular pair of solutions.
Equation ( imply, for example, that the flow of a chemical species i is dependent not only on its conjugate force Xi’ i.e., the difference or negative gradient of its own chemical or electrochemical potentials but also on the gradients or differences of the electrochemical potentials of the other permeating species. Hence eq. ( imply that a difference of electrical potential may cause a flow of an uncharged species, a fact which, as previously indicated, the Nernst-Planck equations do not recognize according to Meares et al In general, eq. ( allow that any type of vectorial force can, under suitable conditions, give rise to any type of vectorial flow.
In a system where n flows are occurring and n forces are operating, a total of n2 phenomenological coefficients Ljk are required to describe fully the transport properties of the system. This must be compared with the n mobilities used in the Nernst-Planck description of the system. A corresponding number n2 experimental transport measurements would have to be made to permit the evaluation of all the Ljk coefficients.
Fortunately a simplification can be made with the help of Onsager’s reciprocal relationship(13). This states that under certain conditions
The conditions required for eq. ( to be valid are that the flows be linearly related to the forces and that the flows and forces be chosen such that where a is the local rate of production of entropy in the system when the Xi are the local potential gradients. The quantity T a is often represented by the symbol ~ and called the dissipation function because it represents the rate at which free energy is dissipated by the irreversible processes. In fact there is no completely general proof of eq. ( but its validity has been shown for a large number of situations(14).
With the help of the reciprocal relationship the number of separate Lik coefficients required to describe a system of n flows and n forces is reduced from n2 to %n (n + 1).
This nonequilibrium thermodynamic theory holds only close to thermodynamic equilibrium. The size of the departure from equilibrium for which the linear relationship between flow and force, eq. (, and the reciprocal relationship, eq. (, are valid, depends upon the type of flow considered. Strictly, the range of validity must be tested experimentally for each type of flow process. In the case of molecular flow processes, electronic conduction, and heat conduction the linear and reciprocal relationships have been found to be valid for flows of the order of magnitude commonly encountered in membranes(65). In describing the progress of chemical reactions the relationships are valid only very close to equilibrium. Systems in which chemical reactions are taking place will be excluded from this discussion.
Although the set of flows and conjugate forces outlined above may seem to be convenient for the molecular interpretation of the interactions occurring in a membrane system, the equations written in terms of these flows and forces are not convenient for the design of experiments for the evaluation of the Lik coefficients. For example, the forces which are usually controlled experimentally are not differences of electrochemical potential, but differences of concentration, electrical potential, and hydrostatic pressure. Also, it may be more convenient to measure the total volume of the flows across a membrane rather than the flow of solvent, or to measure the electric current and one ionic flow rather than two ionic flows. For these reasons, sets of practical flows and forces are often chosen to describe membrane transport(14). These practical sets of flows and their conjugate forces must satisfy the relationship of eq. (, which gives the dissipation function.
A system involving the transport of water and a nonelectrolyte solute across a membrane can be described by giving the flows of water Jw and of solute Js‘ The conjugate forces are then the differences, or the local gradients, of the chemical potentials of water J.Lw and solute J.Ls’ The transport properties of this system are described by the following equations:
When considering ideal external solutions the forces /1J.Lwand /1J.Ls are often expanded into separate terms giving the contributions of the concentration differences and pressure difference to the total driving forces.


2.1 Electro-Osmotic Pumping of Salt Solutions with Homogeneous lon-ExchangeMembranes
2.2 Electro-Osmotic Pumping of Saline Solutions in a Unit-CellStack
2.3 Electro-Osmotic and Osmotic Flows
2.4 Structural Properties of Membranelonomers
2.5 Measurementof Transport Number
2.6 Transport Properties of Anion-ExchangeMembranes in Contact with Hydrochloric Acid Solutions. Membranes for Acid Recovery by ElectrodialysiS
2.7 ElectrodialysisApplications
3.1 Theories of MembraneTransport
3.2 Conductance and Transport Number
3.3 Ion Coupling from ConventionalTransport Coefficients
3.4 Transport Processes Occurring During Electrodialysis
3.5 Current Efficiency and Transport Phenomenain Systems with Charged Membranes
3.6 Efficiency of Energy Conversion in Electrodialysis
3.7 Conversion of Osmotic into MechanicalEnergy in Systems with Charged Membranes
3.8 Donnan Exclusion
3.9 RelationshipBetweenTrue and Apparent Transport Numbers
3.10 Electro-Osmotic Pumping – The Stationary State – Brine Concentration and Volume Flow
3.11 Flux Equations, MembranePotentialsand Current Efficiency
3.12 ElectrodialysisTheory
4.1 Electrodialysis Processesand Stacks
4.2 lon-ExchangeMembranes
4.3 Fouling
4.4 Pretreatment
4.5 Post-treatment
4.6 Seawater Desalination
4.7 Brackish Water Desalination for Drinking-Water Purposes
4.8 Energy Consumption
4.9 Treatment of a High Scaling, High TDS Water with EDR
4.10 Brackish Water Desalination for Industrial Purposes
4.11 ED/IX System
4.12 Industrial Wastewater Desalination for Water Reuse, Chemical Recovery and Effluent Volume Reduction
4.13 Other Possible Industrial Applications
4.14 Polarisation.
4.15 Cell Stack
4.16 Process Design
5.1 Membranes
5.2 Membrane Preparation
5.3 Unit-Cell Construction
5.4 Determination of Brine Concentration, Current Efficiency and Water Flow as a Function of Feed Concentration and Current Density
5.5 Determination of Membrane Characteristics
5.6 Determination of Salt and Acid Diffusion Rate through Membranes .,
5.7 Bench-Scale EOP-ED Stack
5.8 Sealed-Cell ED Stack
6.1 Brine Concentration
6.2 Current Efficiency
6.3 Water Flow
6.4 Membrane Permselectivity
6.5 Membrane Characteristics
7.1 Brine Concentration
7.2 Current Efficiency
7.3 Water Flow
7.4 Membrane Permselectivity
7.5 Acid and Salt Diffusion through Membranes
7.6 Membrane Characteristics
8.1 Brine Concentration
8.2 Current Efficiency
8.3 Water Flow
8.4 Membrane Permselectivity
8.5 Membrane Characteristics
9.1 Concentration/Desalination of Sodium Chloride Solutions with lonac MA-3475 and MC-3470 Membranes
9.2 Concentration/Desalination of Hydrochloric Acid Solutions with Selemion AAVand CHV Membranes
9.3 Concentration/Desalination of Caustic Soda Solutions with Selemion AMV and CMV Membranes
10.1 Concentration of Salt Solutions
10.2 Concentration/Desalination of Industrial Effluents
11.1 Requirements for ED Membranes
11.2 Permselectivity with Acids and Bases
11.3 Brine Concentration, Electro-Osmotic and Osmotic Flows
11.4 Discrepancy between Transport Numbers Derived from Potential Measurements and Current Efficiency Actually Obtained
11.5 Current Efficiency and Energy Conversion in ED
11.6 Water Flow, Concentration Gradient and Permselectivity
11.7 Prediction of Brine Concentration
11.8 Membranes for Sodium Chloride, Hydrochloric Acid and Caustic Soda Concentration
11.9 Conventional EOP-ED Stack
11.10 Sealed-Cell Electrodialysis

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