Binary systems fluxes

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

For the case of a binary system with linear adsorption isotherms, very simple formulas can be derived to evaluate the better TMB flow rates [19, 20]. For the linear case, the net fluxes constraints are reduced to only four inequalities, which are assumed to be satisfied by the same margin /3 (/3 > 1) and so ... 

As an acidic oxide, SiOj is resistant to attack by other acidic oxides, but has a tendency towards fluxing by basic oxides. An indication of the likelihood of reaction can be obtained by reference to the appropriate binary phase equilibrium diagram. The lowest temperature for liquid formation in silica-oxide binary systems is shown below ... 

This leaves (n — 1) independent flux balance equations. The rate v can be found straightforwardly for a binary system as the composition of a and 7 are fixed at any temperature. In a ternary system there is a frirther degree of freedom as the number of thermodynamically possible tie-lines between the a and 7 phases is infinitely large. However, each tie-line may be specified uniquely by the chemical potential of one of the three components and thus there are only two unknowns and two equations to solve. The above approach can be generalised for multi-component systems and forms the platform for the DICTRA software package. 

The mass conservation equation only relates concentration variation with flux, and hence cannot be used to solve for the concentration. To describe how the concentrations evolve with time in a nonuniform system, in addition to the mass balance equations, another equation describing how the flux is related to concentration is necessary. This equation is called the constitutive equation. In a binary system, if the phase (diffusion medium) is stable and isotropic, the diffusion equation is based on the constitutive equation of Pick s law ... 

Uphill diffusion occurs in binary systems because, strictly speaking, diffusion brings mass from high chemical potential to low chemical potential (De Groot and Mazur, 1962), or from high activity to low activity. Hence, in a binary system, a more rigorous flux law is (Zhang, 1993) ... 

The simplest system used for diffusional analysis is that of an isothermal, isobaric binary system where the micromolecular solvent (H20) is designated as component 1 and the solute as component 2. Thus, for concentration gradients of these components, we may measure the net flux of solute across an arbitrary plane or boundary due to the relaxation of the concentration gradient. The interdiffusional flux in a binary liquid mixture is commonly described as mutual diffusion. 

Diffusion in a binary system may also be determined by measurement of the intradiffusion coefficient (sometimes referred to as the self-diffusion coefficient), D. In the case of intradiffusion, no net flux of the bulk diffusant occurs the molecules undergo an exchange process. Measurements are usually carried out by using trace amounts of labelled components in a system free of any gradients in the chemical potential. The molecular movement of the solute is governed by frictional interactions between labelled solute and solvent, and labelled solute and unlabelled solute. 

In the ternary system, therefore, the diffusional flux of water is determined by two of the ternary diffusional coefficients. For a binary system, it was shown earlier that the mutual diffusion of solvent and solute is identical and essentially independent of the magnitude of the osmotic pressure gradient across the boundary 30). 

In the preceding section binary systems are discussed for which Eq. (33) gives the sole contribution to the mass flux. In this section the discussion is extended to binary systems in which Eqs. (33) and (36) are both of importance—that is, both ordinary and thermal diffusion are considered. Then the expression for the mass flux A is... 

Introduction Recently, Kitamura and NakamuraW have found that the anomalous gravity darkening occurs in semi-detached binary systems. The exponent of gravity darkening for the secondary components, which is defined by ac = where F is the radiative flux and g is the... 

In a single sublattice crystal (A, B) with a fixed number of lattice sites and a negligible fraction of vacancies, the sum of the fluxes of A and B has to vanish if the number of sites is to be conserved. We just noted that if we formulate the A and B fluxes in the binary system as usual, they will not be equal in opposite directions because of the differing mobilities (bA 4= bB). However, if we have a local production (annihilation) of lattice sites which operates in such a way as to compensate for any differences in the two fluxes by the local lattice shift velocity, vL, we then obtain... 

The vacancy flux and the corresponding lattice shift vanish if bA = bB. In agreement with the irreversible thermodynamics of binary systems i.e., if local equilibrium prevails), there is only one single independent kinetic coefficient, D, necessary for a unique description of the chemical interdiffusion process. Information about individual mobilities and diffusivities can be obtained only from additional knowledge about vL, which must include concepts of the crystal lattice and point defects. 

Di is the composition-dependent intrinsic diffusivity of component i in a chemically inhomogeneous system. In a binary system, it relates the flux of component i to its corresponding concentration gradient via Fick s law in a local C-frame (which is fixed with respect to the local bulk material of the diffusing system) and is moving with a velocity v with respect to the corresponding V-frame. The Di are related to D as indicated. 

D is the composition-dependent interdiffusivity in a chemically inhomogeneous system. In a binary system, it relates the flux of either component 1 or 2 to its corresponding concentration gradient via Fick s law in a V-frame. 

Consider a binary system at an elevated temperature composed of A and B atoms containing a distribution of spherical /0-phase particles of pure B embedded in an A-rich matrix phase, a. The concentration of B atoms in the vicinity of each /0-phase particle has an equilibrium value that increases with decreasing particle radius, as demonstrated in Fig. 15.1. Because of concentration differences, a flux of B atoms from smaller to larger particles develops in the matrix. This flux causes the smaller particles to shrink and the larger particles to grow. 

Application of this model to published data on the iontophoresis of lidocaine [32,78] in the binary system lidocaine-sodium shows a good agreement between experiment and prediction (Figure 14.5). On the other hand, discrepancies have been found for ropinirole [66] in this case, parallel increments in the concentrations of the drug and sodium were undertaken to maintain their molar fractions constant. While Equation 14.8 predicts that the drug flux should remain constant under these circumstances, ropinirole flux actually decreased as the sodium concentration increased. Clearly, further research is required to optimize the form of this model. 

Finally, it is worth reiterating that transport numbers are relatively complex functions of the concentration and mobility of all the ions present in the system. Thus, while the relationship between lidocaine molar fraction in the binary system (lidocaine-sodium) and the drug flux has been well defined [32,78], the results cannot be directly extrapolated to a different anodal composition. That is, the drug flux depends not only on its molar fraction [59], but also on the mobilities of the competing ions [115]. 

In addition to convection, we must recognize that Fick s law applies exactly to only one solute and one solvent, i.e., to a binary system. In general we should write a more complete flux equation like (de Groot et al., 1962 and Katchalsky et al., 1967) ... 

At x < xg>, the reactivity of the A surface towards the B atoms is less than the flux of these atoms across the ApBq layer. Therefore, there are excessive B atoms which may be used in the formation of either other chemical compounds (enriched in component A in comparison with the ApBq compound) of a multiphase binary system or a solid solution of B in A. 

At x< x f2, there is an excess of diffusing A atoms since the reactivity of the B surface towards these atoms is less than their flux across the ApBq layer. The excessive A atoms can be used in the formation of the layers of other chemical compounds of a given binary system enriched in component B in comparison with ApBq, if present on the equilibrium phase diagram. 

The flux of A can be expressed in terms of concentration for binary systems according to Tick s Law (in spherical coordinates) ... 

Thus, the transformed state variables Xi and transformed fluxes J for each i satisfy the flux equations and conservation equations of a binary system (Stewart and Prober 1964) with the same v function (laminar or turbulent) as the multicomponent system and with a material diffusivity Di = c/Xi, giving a Schmidt number fi/pDi = p/p) Xi/c). 

Olir discussion on diffusion will be restricted primarily to binary systems containing only species A and B. We now wish to determine how the molar diffusive flux of a species (i.e., Ja) is related to its concentration gradient. As an aid in the discussion of the transport law that is ordinarily used to describe diffusion, lesll similar laws ftom other trans K)it processes. For example, in conductive heat transfer the constitutive equation relating the heat flux q and the temperature gradient is Fourier s law ... 

Generally, alcohols showed higher separation factors when present in model multicomponent solutions than in binary systems with water. On the other hand, aldehydes showed an opposite trend. The acmal tea aroma mixmre showed a rather different behavior from the model aroma mixmre, probably because of the presence of very large numbers of unknown compounds. Overall, the PDMS membrane with vinyl end groups used by Kanani et al. [20] showed higher separation factors and fluxes for most of the aroma compounds. Pervaporation was found to be an attractive technology. However, as mentioned above the varying selectivities for the different aroma compounds alter the sensory prohle and therefore application of PV for recovery of such mixmres needs careful consideration on a case-by-case basis. 

Eq. (1) is applicable to both pure diffusion and convective transfer in a laminar or turbulent flow. For a binary system, the total molar flux, which takes into account mass transfer by both molecular diffusion and convection because of bulk flow, can be expressed as ... 

Dependent upon the mode of mass diffusion, i.e., diffusion through a nondiffusing layer, counter-diffusion, or equimolar counter-diffusion, the relationship between the mass transfer coefficient and the diffusivity may take different forms. For example, under the condition of equimolar counter diffusion for a binary system, the molar flux in x-direction can be expressed in terms of the diffusivity as ... 

The setting up of the constitutive relation for a binary system is a relatively easy task because, as pointed out earlier, there is only one independent diffusion flux, only one independent composition gradient (driving force) and, therefore, only one independent constant of proportionality (diffusion coefficient). The situation gets quite a bit more complicated when we turn our attention to systems containing more than two components. The simplest multicomponent mixture is one containing three components, a ternary mixture. In a three component mixture the molecules of species 1 collide, not only with the molecules of species 2, but also with the molecules of species 3. The result is that species 1 transfers momentum to species 2 in 1-2 collisions and to species 3 in 1-3 collisions as well. We already know how much momentum is transferred in the 1-2 collisions and all we have to do to complete the force-momentum balance is to add on a term for the transfer of momentum in the 1-3 collisions. Thus,... 

For the binary systems already discussed, we may regard Eq. 3.1.1 as a linear relationship between the independent flux /j and driving force Vxp For a ternary mixture there are two independent fluxes (7i, J2) and two independent driving forces (Vxj, VX2). Thus, assuming a linear relationship between the fluxes and composition gradients, we may write... 

If we compare Eqs. 5.1.14 with the conservation equation (Eq. 5.1.2) for a binary system and the pseudo-Fick s law Eq. 5.1.15, with Eq. 3.1.1 then we can see that from the mathematical point of view these pseudomole fractions and pseudofluxes behave as though they were the corresponding variables of a real binary mixture with diffusion coefficient D-. The fact that the are real, positive, and invariant under changes of reference velocity strengthens the analogy. If the initial and boundary conditions can also be transformed to pseudocompositions and fluxes by the same similarity transformation, the uncoupled equations represent a set of independent binary-type problems, n - 1 in number. Solutions to binary diffusion problems are common in the literature (see, e.g.. Bird et al., 1960 Slattery, 1981 Crank, 1975). Thus, the solution to the corresponding multicomponent problem can be written down immediately in terms of the pseudomole fractions and fluxes. Specifically, if... 

The presence of nonzero cross-coefficients, D y = 0 (/ = j), in the Fick matrix [D] lends to multicomponent systems characteristics quite different from the corresponding binary system. These characteristics are best illustrated by considering a binary system for which the diffusion flux is given by Eq. 3.1.1... 

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