Diffusivity, mass multicomponent 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. 

According to Maxwell s law, the partial pressure gradient in a gas which is diffusing in a two-component mixture is proportional to the product of the molar concentrations of the two components multiplied by its mass transfer velocity relative to that of the second component. Show how this relationship can be adapted to apply to the absorption of a soluble gas from a multicomponent mixture in which the other gases are insoluble and obtain an effective diffusivity for the multicomponent system in terms of the binary diffusion coefficients. 

Here D km represents a mixture-averaged diffusion coefficient for species k relative to the rest of the multicomponent mixture. The species mass-flux vector is given in terms of the mole-fraction gradient as... 

When considering the mass continuity of an individual species in a multicomponent mixture, there can be, and typically is, diffusive transport across the control surfaces and the production or destruction of an individual species by volumetric chemical reaction. Despite the fact that individual species may be transported diffusively across a surface, there can be no net mass that is transported across a surface by diffusion alone. Moreover homogeneous chemical reaction cannot alter the net mass in a control volume. For these reasons the overall mass continuity need not consider the individual species. At the conclusion of this section it is shown that that the overall mass continuity equation can be derived by a summation of all the individual species continuity equations. 

In a multicomponent mixture, the diffusion rate of a component depends not only on its own concentration in the mixture, but also on the concentration of other components. This may lead to coupling and interaction of the mass transfer among various components. Some examples are (192)... 

It is possible to use alternative formulations considering mole fractions rather than mass fractions. For most cases, mass fraction formulations will be adequate. An estimation of the diffusion coefficient (of component k) in a multicomponent mixture Dkm) however, is not straightforward. For mixtures of ideal gases, the diffusion coefficient in a mixture can be estimated as (Hines and Maddox, 1985)... 

There is a large body of literature that deals with the proper definition of the diffusivity used in the intraparticle diffusion-reaction model, especially in multicomponent mixtures found in many practical reaction systems. The reader should consult references, e.g.. Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., John Wiley Sons, New York, 2002 Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993 and Cussler, Diffusion Mass Transfer in Fluid Systems, Cambridge University Press, 1997. 

Corresponding considerations are also valid for the thermal boundary layer in multicomponent mixtures. The energy transport through conduction and diffusion in the direction of the transverse coordinate x is negligible in comparison to that through the boundary layer. The energy equation for the boundary layer follows from (3.97), in which we will presuppose vanishing mass forces k Ki-... 

The equations (3.109), (3.117) or (3.118) and (3.120) for the velocity, thermal and concentration boundary layers show some noticeable similarities. On the left hand side they contain convective terms , which describe the momentum, heat or mass exchange by convection, whilst on the right hand side a diffusive term for the momentum, heat and mass exchange exists. In addition to this the energy equation for multicomponent mixtures (3.118) and the component continuity equation (3.25) also contain terms for the influence of chemical reactions. The remaining expressions for pressure drop in the momentum equation and mass transport in the energy equation for multicomponent mixtures cannot be compared with each other because they describe two completely different physical phenomena. 

An alternative to the complete Maxwell-Stefan model is the Wilke approximate formulation [103]. In this model the diffusion of species s in a multicomponent mixture is written in the form of Tick s law with an effective diffusion coefficient instead of the conventional binary molecular diffusion coefficient. Following the ideas of Wilke [103] we postulate that an equation for the combined mass flux of species s in a multicomponent mixture can be written as ... 

Let (tu) and (j) denote, respectively, the column arrays of independent mass fractions uji,uj2tOJ3. .., uJq-i and independent diffusion fluxes ji,j2,j3,..., jg i. For multicomponent mixtures with no homogeneous chemical reactions the governing species mass balance can be written on the vector form ... 

Mass transfer is one subject that is unique to chemical engineering. Typical mass transfer problems include diffusion out of a polymer to provide controlled release of a medicine, diffusion inside a porous catalyst where a desired reaction occurs, or a large absorption column where one chemical is transferred from the liquid phase to the gas phase (or vice versa). The models of these phenomena involve multicomponent mixtures and create some tough numerical problems. 

The simplest approach is to calculate binary mass-transfer coefficients F.. from the corresponding empirical correlation, substituting the MS diffusivity D. for the Fick diffusivity in the Sc and Sh numbers. The Maxwell-Stefan equations are, then, written in terms of the binary mass-transfer coefficients. For ideal gas multicomponent mixtures and one-dimensional fluxes, they become... 

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... 

The diffusion or transfer of mass of an i-th component in a multicomponent mixture is possible if there is a spatial concentration gradient of this component (ordinary diffusion), a pressure gradient (barodiffusion), a temperature gradient (thermodiffusion), or if there are external forces that act selectively on the considered component (forced diffusion) [2]. Therefore, concentration, pressure, and... 

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. 

A viscous fluid element comprised of the multicomponent mixture of species moves with the flow. The mass velocity of each species i in the mixture thus consists of flow velocity of the mixture v and diffusion velocity , of species i defined as... 

Traditional equilibrium stage models and efficiency approaches are often inadequate for reactive separation processes. In multicomponent mixtures, diffusion interactions can lead to unusual phenomena, and it is even possible to observe mass transport of the component in the direction opposite to its own driving force - the so-called reverse diffusion (Talyor and Krishna, 1993). For multicomponent systems, the stage efficiencies are different for different components and may range from -< to + >. To avoid possible qualitative errors in the parameter estimation, it is necessary to model... 

This equation accounts for the presence of convection, diffusion, and source terms due to the occurrence of chemical reactions and to the presence of an evaporating wall film and liquid spray. The mass diffusion coefficient F is determined assuming the Pick s law of binary diffusion of a single component into a multicomponent mixture. The momenmm equation can be derived from Eq. (17.44) assuming that the quantity represents a vectorial quantity U ... 

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