Solute flux, definition

An analytical expression for the heat flux vector can be derived in a similar manner using the Enskog approach. That is, we introduce the first order approximation of the distribution function from (2.246) into the heat flux definition (2.72) and thereafter substitute the partial solution for flux vector integrand as follows [39] ... 

A decrease in hydrostatic pressure along the fiber due to resistance to substrate solution flow occurs so that at a definite distance from the inlet, say Lc, transmembrane pressure is nil. Fiber-to-shell solution flux from that point on is negative and becomes a shell-to-fiber flux. Neglecting the shell pressure drop, the overall fiber-to-shell ultrafiltration net flow rate can then be obtained upon integration of the flux equation over the length of the fiber from the inlet to Lc, that is ... 

Rejection of a pressure-driven membrane is typically defined as one minus the ratio of permeate concentration over feed concentration, where the permeate concentration is given by the ratio of permeate solute flux over the water flux (Baker, 2004). Consistent to this definition, the rejection of contaminants in FO processes is defined as ... 

Assume that both the initial substances and the products of the electrode reaction are soluble either in the solution or in the electrode. The system will be restricted to two substances whose electrode reaction is described by Eq. (5.2.1). The solution will contain a sufficient concentration of indifferent electrolyte so that migration can be neglected. The surface of the electrode is identified with the reference plane, defined in Section 2.5.1. In this plane a definite amount of the oxidized component, corresponding to the material flux J0x and equivalent to the current density j, is formed or... 

Even then, the formulation of the system of fluxes and driving forces is not yet satisfactory, because of the quantity Acw. Thus, a new definition of the fluxes will be introduced, i.e. the volume flux of the solution, defined by the equation... 

By definition, Lnjl > 0. The sign of Js is determined by the sign of the cross coefficient Lvjr and its absolute value. If Lv <0, the volume flux of the solvent occurs in the direction from more dilute to more concentrated solutions (i.e. in the direction of the osmotic pressure gradient). If LVJt is smaller than then the solute flows in the direction of the drop of... 

The current is zero at equilibrium. Indeed, = 0 is one definition of equilibrium (see Chapter 2). As the potential is shifted away from V equilibrium so the electrode is polarized (cf Section 6.1). We recall that the deviation of the potential from its equilibrium value is termed the overpotential q (as defined by equation (6.1)). The portion of the Tafel graph at extreme overpotentials represents insufficient flux at the electrode in effect, the potential is so extreme that extra charge could flow if sufficient flux were available but, because of solvent viscosity, rate of solution stirring, etc., the flux is simply not large enough for the behaviour to follow the Tafel equation. 

Following from this definition, the diffusive flux of a solute through the solution and solid in the x direction is given by (Tinker and Nye, 2000, Equation 4.17)... 

Equations (3) and (4) are formally identical with the earlier Kubelka s hyperbolic solutions of differential equations for forward and backward fluxes (11), although the Chandrasekhar-Klier and Kubelka s theories start from different sets of assumptions and employ different definitions of constants characterizing the scattering and absorption properties of the medium. In Kubelka s theory, the constants a, b, and Y are related to the Schuster-Kubelka-Munk (SIM) absorption K and scattering S coefficients as... 

Here f is the mass flux density and c(x, t) is the concentration of a solute, continuously distributed in the spatial field x. For this general anisotropic case B is the positive definite diffusion matrix.1... 

Facilitated diffusion has certain general characteristics. As already mentioned, the net flux is toward a lower chemical potential. (According to the usual definition, active transport is in the energetically uphill direction active transport may use the same carriers as those used for facilitated diffusion.) Facilitated diffusion causes fluxes to be larger than those expected for ordinary diffusion. Furthermore, the transporters can exhibit selectivity (Fig. 3-17) that is, they can be specific for certain molecules solute and not bind closely related ones, similar to the properties of enzymes. In addition, carriers in facilitated diffusion become saturated when the external concentration of the solute transported is raised sufficiently, a behavior consistent with Equation 3.28. Finally, because carriers can exhibit competition, the flux density of a solute entering a cell by facilitated diffusion can be reduced when structurally similar molecules are added to the external solution. Such molecules compete for the same sites on the carriers and thereby reduce the binding and the subsequent transfer of the original solute into the cell. 

By definition, UF membranes are freely permeable to inorganic salts and other molecules with MW less than about 1000. Because it is these species that generally create most of the osmotic pressure of solutions, the net osmotic pressure difference across UF membranes is generally quite small and therefore small applied pressures can be used. Because these membranes are more open than RO membranes, there is less necessity to produce very thin membranes in order to achieve high water fluxes. 

The mass flux of a solute can be related to a mass transfer coefficient which gathers both mass transport properties and hydrodynamic conditions of the system (fluid flow and hydrodynamic characteristics of the membrane module). The total amount transferred of a given solute from the feed to the receiving phase can be assumed to be proportional to the concentration difference between both phases and to the interfacial area, defining the proportionality ratio by a mass transfer coefficient. Several types of mass transfer coefficients can be distinguished as a function of the definition of the concentration differences involved. When local concentration differences at a particular position of the membrane module are considered the local mass transfer coefficient is obtained, in contrast to the average mass transfer coefficient [37]. 

Solution Equation (13.13) can be used to calculate the flux on a circular small electrode by summing along z, the effect of elementary rectangular strips. In this case, x contained in the definition of Si must be replaced by x — Xi), which actually corresponds in the local Cartesian frame of reference to the distance from the leading edge of any elementary strip. The position of the leading edge x (2) is a function of R and z as... 

One can actually consider the trapped solution morphology as a functional definition of the asymmetric membranes. It should be emphasized that this viewpoint clearly differentiates asymmetric membranes that have shown the highest reverse osmosis fluxes from membranes with a thin dense layer of normal solid morphology. 

However, without showing all the lengthy details of the method by which the two scalar functions are determined, we briefly sketch the problem definition in which the partial solution (2.247) is used to determine expressions for the viscous-stress tensor o and the heat flux vector q. 

The definition of the mass transfer coefficient is thus based on an oversimplified picture of the actual physics. The mass transfer coefficient concept relies on the hypothesis that the changes in concentrations are limited to two hypothetical stagnant films, one on each side of the stagnant interface. The transfer flux is thus transfering mass between the interface and the well mixed bulk solution. The amount of matter transferred is expected to be proportional to the concentration difference and the interfacial area. The proportionality coefficient, kc, is called the mass transfer coefficient. The mass transfer coefficient is usually defined by the following flux relation ... 

Enforcing stoichiometric, capacity, and thermodynamic constraints simultaneously leads to the definition of a solution space that contains all feasible steady-state flux vectors. Within this set, one can find a particular steady-state metabolic flux vector that optimizes the network behavior toward achieving one or more goals (e g., maximize or minimize the production of certain metabolites). Mathematically speaking, an objective function has to be defined that needs to be minimized or maximized subject to the imposed constraints. Such optimization problems are typically solved via linear programming techniques. 

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