Phenomenological transport relationships

Phenomenological transport relationships can be developed even in the absence of any knowledge of the mechanisms of transport through the membrane or any information about the membrane structure.10 The basis of irreversible thermodynamics assumes that if the system is divided into small enough subsystems in which local equilibrium exists, thermodynamic equations can be written for the subsystems. 

Dialysis transport relations need not start with Eickian diffusion they may also be derived by integration of the basic transport equation (7) or from the phenomenological relationships of irreversible thermodynamics (8,9). 

Some of the elements of thermodynamics of irreversible processes were described in Sections 2.1 and 2.3. Consider the system represented by n fluxes of thermodynamic quantities and n driving forces it follows from Eqs (2.1.3) and (2.1.4) that n(n +1) independent experiments are needed for determination of all phenomenological coefficients (e.g. by gradual elimination of all the driving forces except one, by gradual elimination of all the fluxes except one, etc.). Suitable selection of the driving forces restricted by relationship (2.3.4) leads to considerable simplification in the determination of the phenomenological coefficients and thus to a complete description of the transport process. 

In the literature, there are many transport theories describing both salt and water movement across a reverse osmosis membrane. Many theories require specific models but only a few deal with phenomenological equations. Here a brief summary of various theories will be presented showing the relationships between the salt rejection and the volume flux. 

This is identical to the Spiegler-Kedem relationship, Eq. (2), and that of finely-porous membrane model, Eq. (3), with a = r. However, it should be noted that Eq. (28) is derived phenomenologically without any assumptions on the transport mechanism. 

There are general relationships of transport phenomena based on phenomenological theory, i.e., on the correlations between macroscopically measurable quantities. The molecular theories explain the mechanism of transport processes taking into account the molecular structure of the given medium, applying the kinetic-statistical theory of matter. The hydrodynamic theories are also applied especially to describe - convection. 

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

The non-ideality of the electrolyte solution leads to complex relationships between the phenomenological coelScients and the transport parameters measured experimentally. These only become simple in the limit of infinite dilution where L x goes to zero and Xc to unity. For example, under these conditions equation (6.7.47) reduces to... 

To describe the functioning of the lEMs, theory from the field of charged membranes must be adapted for MCDI to describe the voltage-current relationship and the degree of transport of the colons. This implies that (in contrast to most membrane processes) the theory must be made dynamic (time dependent) because it has to include the fact that across the membrane the salt concentrations on either side of the membrane can be very different, and change in time. This means that approximate, phenomenological approaches based on (constant values for) transport (or transference) numbers or permselectivities are inappropriate, and that instead a microscopic theory must be used. An appropriate theory includes as input parameters the membrane ion diffusion coefficient and a membrane charge density X. 

Generally, the generic driving force of high temperature corrosion processes can be separated into (i) transport mechanisms, i.e., solid-state diffusion in most cases, and (ii) thermodynamics of chemical reactions. The commonly used, phenomenological way to treat diffusion processes is the application of a second-order partial differential equation (Pick s second law) formulating a relationship between the derivative of the concentration of a species c after... 

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