Diffusion in Nonideal Fluids

The force acting on species i per unit volume of mixture tending to move the molecules of species i is c RTd, where is related to the relative velocities (m - Wy), by 

The appearance of chemical potential gradients in these equations should not come as a surprise. Equilibrium is defined by equality of chemical potentials and departures from 

Chemical potential gradients are not the easiest of quantities to deal with. For nonideal liquids we may express the driving force in terms of the mole fraction gradients as follows  

The symbol X is used to indicate that the differentiation of In % with respect to mole fraction Xy is to be carried out while keeping constant the mole fractions of all other species except the /ith. The mole fraction of species n must be eliminated using the fact that the x sum to unity. More specifically. 

The evaluation of the T y for liquid mixtures from activity coefficient models is discussed at length in Appendix D. 

---

Diffusion is the intermingling of the atoms or molecules of more than one species it is the inevitable result of the random motions of the individual molecules that are distributed throughout space. The development of a rigorous kinetic theory to describe this intermingling in gas mixtures is one of the major scientific achievements of the nineteenth century. A simplified kinetic theory of diffusion, adapted from Present (1958), is the main theme of Section 2.1. More rigorous (and complicated) developments are to be found in the books by Hirschfelder et al. (1964), Chapman and Cowling (1970), and Cunningham and Williams (1980). An extension to cover diffusion in nonideal fluids is developed thereafter. 

With the help of Eq. 2.2.4, develop a more practical expression for the effective diffusivity in nonideal fluids defined in terms of the gradient of mole fraction. 

An exact solution of the Maxwell-Stefan equations for diffusion in nonideal fluids may be obtained using, as a basis, the method developed in Section 8.3.5. All of the results given in that section are valid with the proviso that the matrix [ >] is given by... 

Conversely, the correct approach to formulate the diffusion of a single component in a zeolite membrane is to use the MaxweU-Stefan (M-S) framework for diffusion in a nonideal binary fluid mixture made up of species 1 and 2 where 1 and 2 stands for the gas and the zeohtic material, respectively. In the M-S theory it is recognized that to effect relative motions between the species 1 and 2 in a fluid mixture, a force must be exerted on each species. This driving force is the chemical potential gradient, determined at constant temperature and pressure conditions [68]. The M-S diffiisivity depends on coverage and fugacity, and, therefore, is referred to as the corrected diffiisivity because the coefficient is corrected by a thermodynamic correction factor, which can be determined from the sorption isotherm. 

The linearized theory of Toor (1964) and of Stewart and Prober (1964) discussed in Section 8.4 can be extended to nonideal fluids simply by using the appropriate relation for the matrix of multicomponent diffusion coefficients. For nonideal mixtures the matrix [ )] is evaluated as... 

To illustrate the application of the film model for nonideal fluid mixtures we consider steady-state diffusion in the system glycerol(l)-water(2)-acetone(3). This system is partially miscible (see Krishna et al., 1985). Determine the fluxes Ap A2, and A3 in the glycerol-rich phase if the bulk liquid composition is... 

Using the Maxwell-Stefan equations for nonideal fluids, Eqs. 2.2.1, as a basis, develop a general expression for an effective diffusivity defined in terms of the generalized driving force as... 

For nonideal fluids the driving force for diffusion must be defined in terms of chemical potential gradients as... 

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. 

Third, a serious need exists for a data base containing transport properties of complex fluids, analogous to thermodynamic data for nonideal molecular systems. Most measurements of viscosities, pressure drops, etc. have little value beyond the specific conditions of the experiment because of inadequate characterization at the microscopic level. In fact, for many polydisperse or multicomponent systems sufficient characterization is not presently possible. Hence, the effort probably should begin with model materials, akin to the measurement of viscometric functions [27] and diffusion coefficients [28] for polymers of precisely tailored molecular structure. Then correlations between the transport and thermodynamic properties and key microstructural parameters, e.g., size, shape, concentration, and characteristics of interactions, could be developed through enlightened dimensional analysis or asymptotic solutions. These data would facilitate systematic... 

In real applications, the permeating fluid is a kind dilute solution rather than pure water due to the less-than-ideal membranes with a trace of certain soluble components also diffusing across. The nonideal permeating property of a membrane is reflected by a parameter of rejection which is expressed as... 

Budich, M.. Heilig. S., Wes.se, Th., Leibkiichler, V., Brunner. G. (1999) Countercurrent Deterpenation of Citnis Oils with Supercritical CO2.. lounial (f Supercritical Fluids 14, 105 - 114. van Deemter, J.J., Zuiderweg, F.J., Klinkenberg, A. (1965) Longitudinal diffusion and resistance to mass transfer as causes of nonideality in chromatography. Client Eng Sci 5, 271 - 289. 

At present, computational fluid dynamics methods are finding many new and diverse applications in bioengineering and biomimetics. For example, CFD techniques can be used to predict (1) velocity and stress distribution maps in complex reactor performance studies as well as in vascular and bronchial models (2) strength of adhesion and dynamics of detachment for mammalian cells (3) transport properties for nonhomogeneous materials and nonideal interfaces (4) multicomponent diffusion rates using the Maxwell-Stefan transport model, as opposed to the limited traditional Fickian approach. 

---