Solute flux density

Our final objective in this chapter is to obtain an expression for the solute flux density, Js> that takes into consideration the coupling of forces and fluxes introduced by irreversible thermodynamics. Using Equations 3.34b and 3.35, we note that... 

Equation 3.48 indicates that not only does Js depend on An, as expected from classical thermodynamics, but also that the solute flux density can be affected by the overall volume flux density, Jv. In particular, the classical expression for Js for a neutral solute is P Ac (Eq. 1.8), which equals (Pj/RT)ATlj using the Van t Hoff relation (Eq. 2.10 II, = PT Cj). Thus to, is analogous to P/RT of the classical thermodynamic description (Fig. 3-19). The classical treatment indicates that Js is zero if An is zero. On the other hand, when An is zero, Equation 3.48 indicates that Js is then equal to c,(l - cr,)/y solute molecules are thus dragged across the membrane by the moving solvent, leading to a solute flux density proportional to the local solute concentration and to the deviation of the reflection coefficient from 1. Hence, Pj may not always be an adequate parameter by which to describe the flux of species , because the interdependence of forces and fluxes introduced by irreversible thermodynamics indicates that water and solute flow can interact with respect to solute movement across membranes. 

Soil texture Percentage sand, silt, and clay sized particles contained in the soil. Solute flux density The mass flow rate of solute across a cross-section per unit... 

Here J and J are the transmembrane fluid and solute flux densities Xp and Att are the hydrostatic and osmotic pressure differences (inside minus outside) across the fiber wall C is the "wall" concentration i.e., solute concentration ImmeSiately adjacent to the solution-membrane interface within the fiber annulus C is local filtrate or product concentration at the outer surfice of the fiber and 8 is the transmembrane Pdcldt... 

For very selective membranes 6i grad P [Pg.539]

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

Finally, although the basic development and nature of boiler water plant problems may be similar to those arising in other types of water systems (such as cooling water systems), the extremely high steam-water temperatures and heat-flux densities generally encountered impart a much higher level of intensity. This in turn creates the need for more highly focused and effective solutions to boiler plant problems. 

In electrochemical systems with flat electrodes, all fluxes within the diffusion layers are always linear (one-dimensional) and the concentration gradient grad Cj can be written as dCfldx. For electrodes of different shape (e.g., cylindrical), linearity will be retained when thickness 5 is markedly smaller than the radius of surface curvature. When the flux is linear, the flux density under steady-state conditions must be constant along the entire path (throughout the layer of thickness 8). In this the concentration gradient is also constant within the limits of the layer diffusion layer 5 and can be described in terms of finite differences as dcjidx = Ac /8, where for reactants, Acj = Cyj - c j (diffusion from the bulk of the solution toward the electrode s surface), and for reaction products, Acj = Cg j— Cyj (diffusion in the opposite direction). Thus, the equation for the diffusion flux becomes... 

The pore wall consists of metal oxides. Its reactive part is consumed in proportion to depth because of metabolic activity within a biofilm. The biofilm separates the bulk solution from the pore wall. The horizontal biofilm concentration profiles on the left side of Figure 6 correspond to the center of each of the five boxes. (Concentration is on the vertical axis and biofilm thickness is on the horizontal axis.) Excess organic matter is assumed within the upper few centimeters. The arrows indicate net flux densities of various substances in and out of the biofilm. 

ED appears to be an inefficient method to recover free citric acid because of its low electric conductivity (Novalic et al., 1995). As it is converted into the monovalent (at pH ca. 3), divalent (at pH ca. 5), or trivalent (at pH about 7) citrate anion, there is a significant increase in the electric conductivity (%), the latter increasing from 0.95 to 2.18 and to 3.9 S/m, respectively, in the case of an aqueous solution containing 50 kg/m3 of citric acid equivalent (Moresi and Sappino, 1998). By increasing the pH from 3 to 7, e reduced about eight times, the solute flux (JB) practically doubled, while the overall water transport (/w) increased 3-4 times. The latter partly counterbalanced the greater effectiveness of the electrodialytic concentration of citric acid at pH 7 with respect to that at pH 3. Table XV presents a summary of the effect of current density ( j) on the main performance indicators of the electrodialytic recovery of the monovalent, divalent, or trivalent ionic fractions of citric acid (Moresi and Sappino, 1998). All the mean values or empirical correlations of the earlier indicators were useful to evaluate the economic feasibility of this separation technique (Moresi and Sappino, 2000). 

The ultimate goal of a basic study of separations is to obtain a description of how component concentration pulses (zones or peaks) move around in relationship to one another. The flux density J tells how solute moves across boundaries into and out of regions, but it does not detail the ebb and flow of concentration. To do the latter we must transform J into a form that directly yields concentration changes. The procedure followed below for this is standard in many fields. It is followed, for example, in treatments of heat conduction and diffusion [14,15]. We shall continue to simplify our treatment to one dimension. 

In the filtration-type methods (the first three techniques listed above), components accumulate as a steady-state (polarization) layer at a barrier or membrane [4] this occurs in much the same way as in field-flow fractionation or equilibrium sedimentation. However, there are several complications. First, fresh solute is constantly brought into the layer by the flow of liquid toward and through the filter. This steady influx of solute components can be described by a finite flux density term J0. Second, components can be removed from the outer reaches of the layer by stirring. Third, the membrane or barrier may be leaky and thus allow the transmission of a portion of the solute, profoundly affecting the attempted separation. In fact, one reason for our interest in layer structure is that leakiness depends on the magnitude of the solute buildup at the membrane surface. As solute concentration at the surface increases, more solute partitions into the membrane and is carried on through by flow. 

Since thermal diffusion is a nonisothermal process and thus cannot be considered as driven by chemical potential gradients, we must go directly to the solute flux equations to understand the capacity of thermal diffusion for separation. The basic law expressing the flux density caused by thermal diffusion [46-48] is... 

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... 

For the concentration profile of a sample which has been driven towards the accumulation wall by the physical field, the general transport theory yields for the flux density Jx of the solute ... 

Figure 1-6. Diagram showing the dimensions and the flux densities that form the geometric basis for the continuity equation. The same general figure is used to discuss water flow in Chapter 2 (Section 2.4F) and solute flow in Chapter 3 (Section 3.3A).

Instead of calculating the flux density of some species diffusing into a cell — the amount entering per unit area per unit time — we often focus on the tot al amount of that species diffusing in over a certain time interval. Let sy be the amount of solute species/ inside the cell, where sy can be expressed in moles. If that substance is not involved in any other reaction, dsjldt represents the rate of entry of species / into the cell. The flux density 7/ is the rate of entry of the substance per unit area or (1IA) (ds/dt), where A is the area of the cellular membrane across which the substance is diffusing. Using this expression for Jp we can re-express Equation 1.8 as follows ... 

D. What is the water flux density when the external solution is in equilibrium with a gas phase at 97% relative humidity ... 

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