Flux equations

There are three basic concepts that explain membrane phenomena the Nemst-Planck flux equation, the theory of absolute reaction rate processes, and the principle of irreversible thermodynamics. Explanations based on the theory of absolute reaction rate processes provide similar equations to those of the Nemst-Planck flux equation. The Nemst-Planck flux equation is based on the hypothesis that cations and anions independently migrate in the solution and membrane matrix. However, interaction among different ions and solvent is considered in irreversible thermodynamics. Consequently, an explanation of membrane phenomena based on irreversible thermodynamics is thought to be more reasonable. Nonequilibrium thermodynamics in membrane systems is covered in excellent books1 and reviews,2 to which the reader is referred. The present book aims to explain not theory but practical aspects, such as preparation, modification and application, of ion exchange membranes. In this chapter, a theoretical explanation of only the basic properties of ion exchange membranes is given.3,4 

The Nemst-Planck flux equation has been widely applied to explain transport phenomena in ion exchange membranes and solution systems. When ion i diffuses 

When there is an electrical potential gradient, including diffusion potential, the flux of z, Ji(e), is proportional to the gradient of the electrical potential, (dW/dx), the concentration, C and valence, zi of ion i and its electrochemical mobility w  

This is known as the Nemst-Planck flux equation and is applicable to ideal systems. 

Ion exchange membranes have a large number of hydrated counter-ions in the membrane phase. The counter-ions impart more momentum to the solvent than co-ions do and solvent transfer takes place to the respective electrode chamber 

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To find the total reaction flux, equation (A3.12.12) must be integrated between the limits equal to 0 and E -E, so that... 

Let us now turn attention to situations in which the flux equations can be replaced by simpler limiting forms. Consider first the limiting case of dilute solutions where one species, present in considerable excess, is regarded as a solvent and the remaining species as solutes. This is the simplest Limiting case, since it does not involve any examination of the relative behavior of the permeability and the bulk and Knudsen diffusion coefficients. 

The limiting form of the flux equations for large pore diameters or high pressure is best approached starting from equations (5.7) and (5.8). 

There is a further simplification which is often justifiable, but not by consideration of the flux equations above. The nature of many problems is such that, when the permeability becomes large, pressure gradients become very small ialuci uidiii iiux.es oecoming very large. in catalyst pellets, tor example, reaction rates limit Che attainable values of the fluxes, and it then follows from equation (5,19) that grad p - 0 as . But then the... 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

These are conditions which could be satisfied, even approximately, only in a mixture of isomers of rather similar structure. Then the general dusty gas flux equations (5.4) reduce to... 

The existence of these aimple algebraic relations enormously simplifies the problem of solving the implicit flux equations, since (11.3) permit all the flux vectors to be expressed In terms of any one of them. From equations (11.1), clearly... 

The above estimates of pressure variations suggest that their magni-tude as a percentage of the absolute pressure may not be very large except near the limit of Knudsen diffusion. But in porous catalysts, as we have seen, the diffusion processes to be modeled often lie in the Intermediate range between Knudsen streaming and bulk diffusion control. It is therefore tempting to try to simplify the flux equations in such a way as to... 

Hite s treatment is based on equations (5.18) and (5.19) which describe the dusty gas model at the limit of bulk diffusion control and high permeability. Since temperature Is assumed constant, partial pressures are proportional to concentrations, and it is convenient to replace p by cRT, when the flux equations become... 

Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometiy is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having bulk-flow ports and ports or interfaces through which diffusive flux can occur ... 

Law Simplified flux equations that arise from Eqs. (5-181) and (5-182) can be used for nnidimensional, steady-state problems with binary mixtures. The boundary conditions represent the compositions and I Aft at the left-hand and right-hand sides of a hypothetical layer having thickness Az. The principal restric tion of the following equations is that the concentration and diffnsivity are assumed to be constant. As written, the flux is positive from left to right, as depic ted in Fig. 5-25. 

Stefan-Maxwell Equations Following Eq. (5-182), a simple and intuitively appeahng flux equation for apphcations involving N components is... 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

In a particle having a bidispersed pore structure comprising spherical adsorptive subparticles of radius forming a macroporous aggregate, separate flux equations can be written for the macroporous network in terms of Eq. (16-64) and for the subparticles themselves in terms of Eq. (16-70) if solid diffusion occurs. 

The flux equation assumes constant temperature. As T rises, H rises slowly, but around 25°C the viscosity of water drops enough to produce about a 3 percent rise in flux per °C. 

Equations (22-86) and (22-89) are the turbulent- and laminar-flow flux equations for the pressure-independent portion of the ultrafiltra-tion operating curve. They assume complete retention of solute. Appropriate values of diffusivity and kinematic viscosity are rarely known, so an a priori solution of the equations isn t usually possible. Interpolation, extrapolation, even precuction of an operating cui ve may be done from limited data. For turbulent flow over an unfouled membrane of a solution containing no particulates, the exponent on Q is usually 0.8. Fouhng reduces the exponent and particulates can increase the exponent to a value as high as 2. These equations also apply to some cases of reverse osmosis and microfiltration. In the former, the constancy of may not be assumed, and in the latter, D is usually enhanced very significantly by the action of materials not in true solution. 

The Einstein flux equation for surface diffusion in this situation is... 

The analysis of oxidation processes to which diffusion control and interfacial equilibrium applied has been analysed by Wagner (1933) who used the Einstein mobility equation as a starting point. To describe the oxidation for example of nickel to the monoxide NiO, consideration must be given to tire respective fluxes of cations, anions and positive holes. These fluxes must be balanced to preserve local electroneutrality tliroughout the growing oxide. The flux equation for each species includes a term due to a chemical potential gradient plus a term due to the elecuic potential gradient... 

The complete flux equations when the cation has a valence m+, and the anion a valence r— in the compound... 

On substituting diese values into the ionic flux equations we obtain... 

Therefore, for this system the water flux equation (11.9a) is ... 

Application of the exergy flux equation to a closed cycle... 

We next consider the application of the exergy flux equation to a closed cycle plant based on the Joule-Brayton (JB) cycle (see Fig. 1.4), but with irreversible compression and expansion processes—an irreversible Joule-Brayton (IJB) cycle. The T,.s diagram is as shown in Fig. 2.6. 

It is important to note tlmt tlic deposition rate is a strong function of particle dimneter tluough the term v, wliich appears twice in tlic deposition flux equation. Equation (9.7.10) must be modified to treat process gas streams discliarging particles of a given size distribution. The suggested procedure is somewhat simihu to tlial for calculating overall collection efficiencies for particulate control equipment (12). For this condition, the overall rale is given by... 

In the case of systems containing ionic liquids, components and chemical species have to be differentiated. The methanol/[BMIM][PF6] system, for example, consists of two components (methanol and [BMIM][PFg]) but - on the assumption that [BMIM][PFg] is completely dissociated - three chemical species (methanol, [BMIM] and [PFg] ). If [BMIM][PFg] is not completely dissociated, one has a fourth species, the undissociated [BMIM][PFg]. From this it follows that the diffusive transport can be described with three and four flux equations, respectively. The fluxes of [BMIM] ... 

The maximum flux equation of Zuber is suggested as another check for ketde reboilers ... 

Following Sadler [161] (the details were not included in his work) consider a row model in which all the stems except the outermost are of the same length, h, as shown in Fig. 4.9. A new stem can only be initiated when the outermost stem has reached a height h. A set of flux equations may be written as ... 

The value of the integral in the energy balance (equation 11.55) is again given by equation 11.60 [substituting (6S - 8o) for 0 ]. The heat flux q0 at the surface is now constant, and the right-hand side of equation 11.55 may be expressed as (—qa/Cf,p). Thus, for constant surface heat flux, equation 11.55 becomes ... 

The relevance of Eq. (2.2) (which predicts how quickly molecules pass through simple membranes) to solubility comes in the concentration terms. Consider sink conditions, where Ca is essentially zero. Equation (2.2) reduces to the following flux equation... 

Palm et al. [578] derived a two-way flux equation which is equivalent to Eq. (7.13), and applied it to the permeability assessment of alfentanil and cimeti-dine, two drugs that may be transported by passive diffusion, in part, as charged species. We will discuss this apparent violation of the pH partition hypothesis (Section 7.7.7.1). 

When very insoluble samples are used, sometimes precipitate forms in the donor wells, and the solutions remain saturated during the entire permeation assay. Equations (7.20) and (7.21) would not appropriately represent the kinetics. One needs to consider the following modified flux equations [see, Eqs. (7.1) and (7.2)]... 

When the pH is different on the two sides of the membrane, the transport of ioniz-able molecules can be dramatically altered. In effect, sink conditions can be created by pH gradients. Assay improvements can be achieved using such gradients between the donor and acceptor compartments of the permeation cell. A three-compartment diffusion differential equation can be derived that takes into account gradient pH conditions and membrane retention of the drug molecule (which clearly still exists—albeit lessened—in spite of the sink condition created). As before, one begins with two flux equations... 

The term hID is often called the diffusional resistance, denoted by R. The flux equation, therefore, can be written as... 

Control of nutrient transport dictates significant coupling between transported components in G1 epithelia. This complicates solute transport analysis by requiring a multicomponent description. Flux equations written for each component constitute a nonlinear system in which the coupling nonlinearities are embodied in the coefficients modifying individual transport contributions to flux. 

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