Total molar density

To integrate this, u is needed. When there is no change in the number of moles upon reaction. Equation (3.2) applies to the total molar density as well as to the mass density. Thus, for constant A, ... 

In Equation (9.6), x is the direction of flux, nt [mol m-3 s 1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m-2 s 1] is the mole flux due to molecular diffusion, D k [m2 s 1] is the effective Knudsen diffusion coefficient, D [m2 s 1] is the effective bimolecular diffusion coefficient (D = Aye/r), e is the porosity of the electrode, r is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983 Warren, 1969). This model modifies Equation (9.6) as ... 

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... 

U Deposition Uniformity U = fjf = f ) f) Reactant Mean Molecular Speed X Transverse Position z Normalized Transverse Position (z = x/a) a Ratio of Diffusivities P Deposition Modulus A Reactant Mean Free Path Normalized Preform Thickness p Total Molar Density o Reactive Species Molecular Diameter ( ) Reaction Probability / Net Molar Yield of Deposition Reaction... 

Dry air is blown on top of the tube and, thus, r = Xi. i = 0. Also, the total molar density throughout the lube remains constant because of the constant teniperature and pressure conditions and is determined to be... 

For a multicomponent mixture of ideal gases such as this, the Maxwell-Stefan equations (1-35) must be solved simultaneously for the fluxes and the composition profiles. At constant temperature and pressure the total molar density c and the binary diffusion coefficients are constant. Furthermore, diffusion in the Stefan tube takes place in only one direction, up the tube. Therefore, the continuity equation simplifies to equation (1-69). Air (3) diffuses down the tube as the evaporating mixture diffuses up, but because air does not dissolve in the liquid, its flux N3 is zero (i.e., the diffusion flux J3 of the gas down the tube is exactly balanced by the diffusion-induced bulk flux x3N up the tube). 

In this system, we designate water as species A and air is referred to as species B. The mass fractions are denoted by xa and xb and they are defined as the mass of the individual species divided by the sum of the masses of the species. The molar fractions yx and jb equal the moles of species A or B with respect to the total number of moles. Remember that the total molar density equals for gases. Figure 7.12 demonstrates the evolution of the mass fraction of water as a function of time. The first vertical solid line represents the semi-permeable membrane. The fabric extends from the membrane to the dotted line. Initially, the mass fraction of water in the fabric equals zero. At the beginning of the experiment, the mass fraction of water at the membrane equals one and it is zero in the fabric. As time advances, the mass fraction in the fabric increases (third image from the top). With time, the concentration profile of species A eventually becomes a straight line, as shown in Figure 7.12. 

C2J — Cp the total molar density of the mixture in region/ The mass concentration... 

The MaxweU-Stefan equations for describing the diffusion of gases in a multicomponent gas mixture have been developed from the kinetic theory of gases. A highly simplified illustration may be pursued as follows. Consider a system of a gas mixture of n species at constant T and P. Focus first on molecules of species i. The net force exerted on species i in the absence of any external forces is - Vpig / gmol of i. The net force exerted on species i per unit volume of the mixture is —(V/tjg) xig Ctg, where C,g is the total molar density of the gas mixture from equation (3.1.40) and an ideal gas mixture, the net force on species i per unit mixture volume is —xjg Ctg Rr V tn Pxtg), which is equivalent to... 

Assume ideal solution in the membrane and a constant total molar density C,m across the membrane. Then... 

In the derivation of this equation, we assumed that the total molar density Ca of the liquid phase is essentially constant and that the vapor entering plate n from plate [n + 1) below has a uniform composition x,i( + 1). We will now obtain simpler forms of equation (8.3.18) for a variety of situations by making additional assumptions. 

The species continuity equations ensure conservation of atoms by balancing the chemical equations governing the reaction system. The concentration of an individual species i will be expressed either as a product of total molar density and mole fraction X, or as a product of density and mass fraction w,. In an N-component system mass fraction and mole fraction are related by... 

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