Binary systems molecular diffusion

The effective diffusivity De is a characteristic of the particle that must be measured for greatest accuracy. However, in the absence of experimental data, De may be estimated in terms of molecular diffusivity, Dab (for diffusion of A in the binary system A + B), Knudsen diffusivity, DK, particle voidage, p, and a measure of the pore structure called the particle tortuosity, Tp. 

There are a number of quantitative features of Eq. (14) which are important in relation to rapid diffusional transport in binary systems. The mutual diffusion coefficient is primarily dependent on four parameters, namely the frictional coefficient 21 the virial coefficients, molecular weight of component 2 and its concentration. Therefore, for polymers for which water is a good solvent (strongly positive values of the virial coefficients), the magnitude of (D22)v and its concentration dependence will be a compromise between the increasing magnitude of with concentration and the increasing value of the virial expansion with concentration. 

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. 

Consider the problem of steady-state one-dimensional diffusion in a mixture of ideal gases. At constant T and P, the total molar density, c = P/RT is constant. Also, the Maxwell-Stefan diffusion coefficients D m reduce to binary molecular diffusion Dim, which can be estimated from the kinetic theory of gases. Since Dim is composition independent for ideal gas systems, Eq. (6.61) becomes... 

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 ... 

The analysis of turbulent eddy transport in binary systems given above is generalized here for multicomponent systems. The constitutive relation for j y in multicomponent mixtures taking account of the molecular diffusion and turbulent eddy contributions, is given by the matrix generalization of Eq. 10.3.1... 

In reactive flow analysis the Pick s law for binary systems (2.285) is frequently used as an extremely simple attempt to approximate the multicomponent molecular mass fluxes. This method is based on the hypothesis that the pseudo-binary mass flux approximations are fairly accurate for solute gas species in the particular cases when one of the species in the gas is in excess and acts as a solvent. However, this approach is generally not recommend-able for chemical reactor analysis because reactive mixtures are normally not sufficiently dilute. Nevertheless, many industrial reactor systems can be characterized as convection dominated reactive flows thus the Pickian diffusion model predictions might still look acceptable at first, but this interpretation is usually false because in reality the diffusive fluxes are then neglectable compared to the convective fluxes. 

In this equation D is the molecular diffusivity of binary systems, and is the chemical source term for species c. The scalar fields are assumed to be passive scalars so that ujc has no significant influence on the momentum and continuity equations. 

The present review of the theory of transport processes in liquids is confined to considerations of the primary molecular mechanism involved in these processes. Thus the questions of irreversibility of nonequilibrium systems in general and of the reciprocal relations among the transport coefficients in coupled processes are not developed. Also in the interest of brevity, the special problems of binary and thermal diffusion are not dealt with in detail. This being a review of theory, there is no attempt to present the numerous experimental studies of transport processes in various liquid systems in the past decade. 

D is the molecular diffusivity of the nitrogen-helium binary system (in A ft... 

When treating diffusion of solutes in porous materials where diffusion is considered to occur only in the fluid inside the pores, it is common to refer to an effective diffusivity, DABeg, which is based on (1) the total cross-sectional area of the porous solid rather than the cross-sectional area of the pore and (2) on a straight path, rather than the actual pore path, which is usually quite tortuous. In a binary system, if pore diffusion occurs only by ordinary molecular diffusion, Fick s law can be used with an effective diffusivity that can be expressed in terms of the ordinary diffusion coefficient, DAB, as... 

Jj is the molar flux vector for species j with respect to the mass average velocity (kmol/m s). When the flow is laminar or perfectly ordered the term V Jj results from molecular diffusion only. It can be written more explicitly as an extension, already encountered in Chapter 3, of Pick s law for diffusion in binary systems, as... 

MOLECULAR DIFFUSION COEFFICIENTS IN BINARY GASEOUS SYSTEMS AT ONE ATMOSPHERE PRESSURE. N-HEXANE-METHANE AND 3-METHYLPENTANE-METHANE SYSTEMS. 

We have presented the necessary equation to relate flux and mole fraction gradient for a multicomponent system (eqs. 8.6-18) when both molecular diffusion and Knudsen diffusion are operating. Let us now treat a special case of binary systems. For such a case, the Stefan-Maxwell equations are ... 

Pol5mer solutions are binary systems (we assume the polymer is monodis-perse in relative molecular mass) and the variation of surface tension with composition is governed by the Gibbs equation in the same manner as it is for molecules of low relative molecular mass. In principle the hypothetical dividing surface is placed so that each phase either side is uniform up to the surface. In practice, because liquid surfaces are diffuse (due to evaporation processes and capillary waves), the dividing surface is usually placed so that the surface excess of solvent is zero. Figure 8.21 illustrates this and also defines the surface excess of solute. 

Since Pick cast his equation in a familiar form and since Eqs. ri5-2bi and fl5-4ai fit data for isothermal dilute binary systems very well, this equation rapidly became enshrined as Pick s law (sometimes known as Pick s first law). However, problems arose when other researchers extended Pick s work to more concentrated systems. In Section 15.2.3 we will see that when there is significant convection in the diffusion direction, the diffusion flux J needs to be related to the flux N with respect to a fixed coordinate system (N is the flux needed to design equipment). This conplicates the picture but does not invalidate Pick s law. As we shall see later, when extended to concentrated, nonideal systems or to multiconponent systems. Pick s law often requires very large adjustments of the molecular diffusivity—sometimes with negative values—as a function of concentration to predict behavior. Said in clearer terms. Pick s law no longer applies. We should not blame Pick for this lack of agreement. His law works fine for the conditions that he developed it for. 

Equation (7.1-16) reduces to two special cases of molecular diffusion which are customarily considered. In equimolal counterdiffiuion, component A diffuses through component B, which is difliising at the same moM rate as A relative to statimiaiy coordinates, but in the of xisite direction. This process is often approximated in the distillation of a binary system. In unimolal unMrectional diffitsion, only one molecular species—component A—diffuses through component B, which is motionless relative to stationary coordinates. This type of transfer is approximated frequently in the operations of gas absorption, liquid-liquid extraction, and adsorption. 

The study of the concentration dependence / is a very informative and sensitive method for analyzing the molecular interactions in binary systems (Vuks, 1974). This dependence is obtained experimentally from light scattering data in solution or diffusion. Then, >ising / J xi), one can calculate the most important thermodynamic quantities of the system. 

The molecular diffusivity of a binary gas mixture is essentially independent of composition. In multicomponent mixtures the diffusivity becomes, in principle, concentration dependent, but such variations are generally relatively small so that the assumption of a concentration-independent diffusivity is usually a good approximation for most gaseous systems. Other important general conclusions which follow from Eq. (5.15) are that the molecular diffusivity is inversely dependent on total pressure and proportional to a low power of temperature. The combined effect of the factor in the numerator and the temperature-dependent function 2 i/kT) in the denominator yields an overall temperature dependence of approximately T . 

To summarize, the existence of mesoscopic domains in the W/bmimBF binary system can be inferred from the analysis of the self-diffusion coefficients of the various molecular species in solution, since they were found to obey in a different way to a fractional Stokes-Einstein equation. In addition, bmim+ and BF self-diffusion measurements, although suggesting some form of association of the cations, clearly evidenced that micellar aggregates did not form at any composition. 

The use of ANN is highly developed due their great advantage compared with traditional computing systems. ANNs have a flexible structiue, capable to make a nonlinear mapping between input and output data sets. In fact, multilayer perceptrons, one of the more extended neural network architectures, are imiversal approximators for complex problems [12]. The apphcation of this is reflected in the hteratiue devoted to prediction of many physical and chemical parameters, such as nanofluids density [14], density of binary mixtures of ionic hquids [15], electrical percolation temperatiue [16], molecular diffusivity of nonelectrolytes [17], vegetable oils viscosity [18], esters flash point prediction [12], polarity parameter in binary mixed solvents systems [19], etc. 

STEADY-STATE MOLECULAR DIFFUSION IN BINARY SYSTEMS... 

The molecular diffusion coefficient, D i, can be calculated fi om the binary diffusion coefficients (Dy) obtained from the Fuller-Schettler-Giddings equation [20] for gas-phase systems. The individual diffusion coefficients, Dmi, can be estimated from the binary diffusion coefficients using Wilke s approximation [20]. For the Knudsen diffusion... 

In simple molecular diffusion in a binary system under iso-baric condition. 

In this equation D is the molecular diffusivity of binary systems, and Su>c is the chemical source term for species c. The scalar fields are assumed to be passive scalars so that ujc has no significant influence on the momentum and continuity equations. After suitable non-dimensional variables are substituted into theequations, following the same procedure as outlined in Sect. 1.2.5, the important dimensionless groups are obtained for the problem in question. These are the Reynolds number, the Schmidt number, the Peclet number, Pe = Re Sc = ul/D, and the Damkohler number, Dui = Ir/u. The u and I are the characteristic velocity and length scales, respectively, for the velocity field, and r denotes a characteristic chemical reaction rate. 

If a gas or liquid mixture is ideal, then Da/ is the binary molecular diffiisivity of A in species i . Binary molecular diffusivities are almost independent of concentration for ideal gas and liquid systems. Values of Dy for many common binary pairs are tabulated in handbooks, and methods for predicting binary diffusivities as a function of temperature and pressure are... 

---