Multicomponent Diffusion in a Fluid

For a binary mixture, the single diffusivity Dab is used in the Schmidt number. Most practical problems involve multicomponent mixtures, whose rigorous treatment is much more complicated, however. 

In general, the flux of a given chemical species can be driven not only by its own concentration gradient, but also by those of all the other species [see Toor (1964), for example]  

The form of (3.2.3-1) is often judged to be too complex for many engineering calculations. A common approach is to define a mean binary diffusivity for species j diffusing through the mixture  

Using (3.2.3-1), Toor [1964] and Stewart and Prober [1964] showed that the matrix of the Djt could be diagonalized, which then gives the form of (3.2.3-2), and the many solutions available for binary mixtures can be adapted to multicomponent mixtures. 

The Stefan-Maxwell equations for ideal gases are given in Bird, Stewart, and Lightfoot [1960] as  

The last term accounts for bulk flow of the mixture. The exact form of the depends on the system under study. For ideal gases, the kinetic theory leads to the Stefan-Maxwell equations, which can be rearranged into the form of Eq. 

c-l-a treatment using matrix methods is given by Stewart and Prober [18]. 

For liquids, there is no complete theory yet available—for a discussion of corrections for thermodynamic nonidealities, and other matters, see Bird, Stewart, and Lightfoot [2]. A comprehensive review of available information on gas diffusion is by Mason and Marrero [19], and for liquids sec Dullien, Ghai, and Ertl 

---

It is not known whether high-pressure fluids are Newtonian fluids that behave according to the laws given by Eqns. (3.4-1), (3.4-2), and (3.4-3). With regards to diffusion problems, for example, the Fickian nature of diffusion may be rather the exception than the rule. The diffusivity often depends on solute concentration, not only in extraction with a supercritical gas [1] but also in ordinary low-pressure diffusion in the gas phase and in diffusion in a liquid in multicomponent systems and in porous media. 

A. Pfennig, Multicomponent Diffusion, in Int. Workshop "Transport in Fluid Multiphase Systems From Experimental Data to Mechanistic Models. 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

The transfer of heat in a fluid may be brought about by conduction, convection, diffusion, and radiation. In this section we shall consider the transfer of heat in fluids by conduction alone. The transfer of heat by convection does not give rise to any new transport property. It is discussed in Section 3.2 in connection with the equations of change and, in particular, in connection with the energy transport in a system resulting from work and heat added to the fluid system. Heat transfer can also take place because of the interdiffusion of various species. As with convection this phenomenon does not introduce any new transport property. It is present only in mixtures of fluids and is therefore properly discussed in connection with mass diffusion in multicomponent mixtures. The transport of heat by radiation may be ascribed to a photon gas, and a close analogy exists between such radiative transfer processes and molecular transport of heat, particularly in optically dense media. However, our primary concern is with liquid flows, so we do not consider radiative transfer because of its limited role in such systems. 

The parameter D is known as the axial dispersion coefficient, and the dimensionless number, Pe = uL/D, is the axial Peclet number. It is different than the Peclet number used in Section 9.1. Also, recall that the tube diameter is denoted by df. At high Reynolds numbers, D depends solely on fluctuating velocities in the axial direction. These fluctuating axial velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for D is used for each component in a multicomponent system. 

Klimenko, A. Y. (1990). Multicomponent diffusion of various admixtures in turbulent flow. Fluid Dynamics 25, 327-334. 

Most diffusion processes encountered in Earth sciences are, strictly speaking, multicomponent diffusion. For example, even "self "-diffusion of oxygen isotopes from an 0-enriched hydrothermal fluid into a mineral is likely due to chemical diffusion of H2O into the mineral (see Section 3.3.3). Because a natural melt contains at least five major components and many trace components, diffusion in nature is complicated to treat. For multicomponent and anisotropic minerals,... 

According to their analysis, if c is zero (practically much lower than 1), then the fluid-film diffusion controls the process rate, while if ( is infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the mechanical parameter represents the ratio of the diffusion resistances (solid and fluid-film). This equation can be used irrespective of the constant pattern assumption and only if safe data exist for the solid diffusion and the fluid mass transfer coefficients. In multicomponent solutions, the use of models is extremely difficult as numerous data are required, one of them being the equilibrium isotherms, which is a time-consuming experimental work. The mathematical complexity and/or the need to know multiparameters from separate experiments in all the diffusion models makes them rather inconvenient for practical use (Juang et al, 2003). 

This simplification is not possible for some CVD systems in which large density changes are associated with the deposition process. The growth of CdHgTe is a typical example that shows how the depletion of Hg next to the substrate creates an unstable density gradient that drives recirculations (205), as discussed earlier and illustrated in Figure 14. LPCVD processes use little or no diluent and often involve several species, and multicomponent diffusion may be an important factor (21). Fortunately, these reactors are isothermal, and the relative insensitivity of reactor performance to details of the fluid flow greatly simplifies the analysis. 

Solute-solute Interactions may affect the diffusion rates In the fluid phase, the solid phase, or both. Toor (26) has used the Stefan-Maxwell equations for steady state mass transfer In multicomponent systems to show that, in the extreme, four different types of diffusion may occur (1) diffusion barrier, where the rate of diffusion of a component Is zero even though Its gradient Is not zero (2) osmotic diffusion, where the diffusion rate of a component Is not zero even though the gradient Is zero (3) reverse diffusion, where diffusion occurs against the concentration gradient and, (4) normal diffusion, where diffusion occurs In the direction of the gradient. While such extreme effects are not apparent in this system, it is evident that the adsorption rate of phenol is decreased by dodecyl benzene sulfonate, and that of dodecyl benzene sulfonate increased by phenol. 

The principles and basic equations of continuous models have already been introduced in Section 6.2.2. These are based on the well known conservation laws for mass and energy. The diffusion inside the pores is usually described in these models by the Fickian laws or by the theory of multicomponent diffusion (Stefan-Maxwell). However, these approaches basically apply to the mass transport inside the macropores, where the necessary assumption of a continuous fluid phase essentially holds. In contrast, in the microporous case, where the pore size is close to the range of molecular dimensions, only a few molecules will be present within the cross-section of a pore, a fact which poses some doubt on whether the assumption of a continuous phase will be valid. 

The optimal Reynolds number defines the operating conditions at which the cylindrical system performs a required heat and mass transport, and generates the minimum entropy. These expressions offer a thermodynamically optimum design. Some expressions for the entropy production in a multicomponent fluid take into account the coupling effects between heat and mass transfers. The resulting diffusion fluxes obey generalized Stefan-Maxwell relations including the effects of ordinary, forced, pressure, and thermal diffusion. 

For binary diffusion, there is only one independent flow, force or concentration gradient, and diffusion coefficient. On the other hand, multicomponent diffusion differs from binary diffusion because of the possibility of interactions among the species in mixtures of three or more species. Some of the possible interactions are (1) a flow may be zero although its zero driving force vanishes, which is known as the diffusion barrier (2) the flow of a species may be in a direction opposite to that indicated by its driving force, which is called reverse flow and (3) the flow of a species may occur in the absence of a driving force, which may be called osmotic flow. The theory of nonequilibrium thermodynamics indicates that the chemical potential arises as the proper driving force for diffusion. This is also consistent with the condition of fluid phase equilibrium, which is satisfied when the chemical potentials of a species are equal in each phase. 

Here, Dy is an empirical, radial dispersion coefficient and e is the void fraction. The units of diffusivity Dy are square meters per second. The major differences between this model and the convective diffusion equation used in Chapter 8 is that the velocity profile is now assumed to be flat and Dy is an empirically determined parameter instead of a molecular diffusivity. The value of Dy depends on factors such as the ratio of tube to packing diameters, the Reynolds number, and (at least at low Reynolds numbers) the physical properties of the fluid. Ordinarily, the same value for Dy is used for all reactants, finessing the problems of multicomponent diffusion and allowing the use of stoichiometry to eliminate Equation 9.1 for some of the components. Note that Us in Equation 9.1 is the superficial velocity, this being the average velocity that would exist if the tube had no packing. 

Answer The product of Re and Sc is the mass transfer Peclet number, Pcmt, where the important mass transfer rate processes are convection and diffusion. Since the dimensional scaling factors for both of these rate processes do not contain information about the constitutive relation between viscous stress and velocity gradients, one concludes that PeMT is the same for Newtonian and non-Newtonian fluids. Hence, the mass transfer Peclet number for species in a multicomponent mixture is... 

Diffusion is the mass transfer caused by molecular movement, while convection is the mass transfer caused by bulk movement of mass. Large diffusion rates often cause convection. Because mass transfer can become intricate, at least five different analysis techniques have been developed to analyze it. Since they all look at the same phenomena, their ultimate predictions of the mass-transfer rates and the concentration profiles should be similar. However, each of the five has its place they are useful in different situations and for different purposes. We start in Section 15.1 with a nonmathematical molecular picture of mass transfer (the first model) that is useful to understand the basic concepts, and a more detailed model based on the kinetic theory of gases is presented in Section 15.7.1. For robust correlation of mass-transfer rates with different materials, we need a parameter, the diffusivity that is a fundamental measure of the ability of solutes to transfer in different fluids or solids. To define and measure this parameter, we need a model for mass transfer. In Section 15.2. we discuss the second model, the Fickian model, which is the most common diffusion model. This is the diffusivity model usually discussed in chemical engineering courses. Typical values and correlations for the Fickian diffusivity are discussed in Section 15.3. Fickian diffusivity is convenient for binary mass transfer but has limitations for nonideal systems and for multicomponent mass transfer. 

The model of Santacesaria et is an extension of the linear driving force model, with fluid side resistance, for a nonlinear multicomponent Langmuir system. It includes axial dispersion, and the combined effects of pore diffusion and external fluid film resistance are accounted for throu an overall rate coefficient. Intracrystalline diffusional resistance is neglected and equilibrium between the fluid in the macfopores and in the zeolite crystals is... 

The conservation of species is actually the law of conservation of mass applied to each species in a mixture of various species. The fluid element, as described in Sections 6.2.1.1 through 6.2.1.3, does not comprise of pure fluid with only one species, such as water, but of many species forming a multicomponent mixture. This law is mathematically described by the continuity equation for species, also known as the species equation, advection-diffusion equation, or convection-diffusion equation. If the species equation additionally includes a reaction term, it is known as the reaction-diffusion-advection equation. 

---