Total molar flux

Now consider Che cross-sectional average N of the total molar flux and the cross-sectional average diffusion fluxes J, defined by... 

A simple treatment is stiU possible if it may be assumed that the flux of the component of interest A through the interface stays in a constant proportion to the total molar transfer through the interface over the entire tower ... 

For an ideal gas, the total molar concentration Cj is constant at a given total pressure P and temperature T. This approximation holds quite well for real gases and vapours, except at high pressures. For a liquid however, CT may show considerable variations as the concentrations of the components change and, in practice, the total mass concentration (density p of the mixture) is much more nearly constant. Thus for a mixture of ethanol and water for example, the mass density will range from about 790 to 1000 kg/m3 whereas the molar density will range from about 17 to 56 kmol/m3. For this reason the diffusion equations are frequently written in the form of a mass flux JA (mass/area x time) and the concentration gradients in terms of mass concentrations, such as cA. 

Transient computations of methane, ethane, and propane gas-jet diffusion flames in Ig and Oy have been performed using the numerical code developed by Katta [30,46], with a detailed reaction mechanism [47,48] (33 species and 112 elementary steps) for these fuels and a simple radiation heat-loss model [49], for the high fuel-flow condition. The results for methane and ethane can be obtained from earlier studies [44,45]. For propane. Figure 8.1.5 shows the calculated flame structure in Ig and Og. The variables on the right half include, velocity vectors (v), isotherms (T), total heat-release rate ( j), and the local equivalence ratio (( locai) while on the left half the total molar flux vectors of atomic hydrogen (M ), oxygen mole fraction oxygen consumption rate... 

Coordinate in axial direction Molecular weight of volatile component Mass of gas in bubble Total molar flux of volatile component... 

Dj is the mass diffusion coefficient, and cgas is the total molar concentration of the gas mixture. Although Equations (3.9a) and (3.9b) can be used for a free-path gas (e.g. gas channel), when a gas is moving within a porous media (i.e. electrode), Equation (3.9) may not be the most appropriate. Different constitutive laws can be employed for describing the diffusive flux within a porous medium. The choice of the most appropriate law depends on the operating conditions and the porous media properties, as further explained in Section 3.3.2. 

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 ... 

Using the diffusion fluxes JU(. the total molar fluxes are defined by... 

Step 4 Estimate the total molar fluxes A, from Eq. (6.78)... 

Step 4 Calculate the total molar fluxes. Due to interface invariance Nx = N,.. we have... 

In these equations cj and cf are the molar densities of the superscripted phases, yj is the mole fraction in the bulk vapor phase, xf is the mole fraction in the bulk liquid phase, and xj and yj are the mole fractions of species i at the phase interface. Also is the total molar flux in phase p, and kj and are the mass-transfer coefficients for... 

The above model has been refined based on the dusty gas model [Mason and Malinauskas, 1983] for transport through the gas phase in the pores and the surface diffusion model [Sloot, 1991] for transport due to surface flow. Instead of Equation (10-101), the following equation gives the total molar flux through the membrane pores which are assumed to be cylindrically shaped... 

In a binary flow system with interfacial fluid composition xao and bulk fluid composition x b, the local mass-transfer coefficient k corrected for the interfacial total molar flux cvq + Nbo] satisfies... 

Using Pick s law of diffusion, the total molar fluxes j — NIA and diffusion molar flosv rates can be expressed as... 

The total molar flux of A is given by Equation (11-1). Ba can be expressed either in terms of the concentration of A, in which case... 

As the molar flux of each of the two components is independent of the position coordinate y, the total molar flux N/A and likewise the quotient Nx/N are also independent of y. Therefore (4.59) can be integrated easily. The integration extends from the condensate surface (index I) to the vapour space (index G). The thickness of the vapour boundary layer will be 5. We assume constant values for the pressure and temperature. Under the assumption that the gas phase exhibits ideal behaviour, the diffusion coefficient and the molar concentration c = N/V = p/(Rm T) are likewise independent of the coordinate y. The integration yields... 

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 ... 

Total molar flux relative to stationary coordinates... 

The diffusion fluxes 7, and the molar fluxes are equal in this case because the total molar flux Ni is zero. 

The composition profiles, calculated from Eq. 8.4.3 are shown in Figure 8.9. Notice the quite large change in the mole fraction of ethanol. Despite this large change, the profiles themselves are not highly curved this being the result of the relatively low total molar flux. 

The classic result Sh = 2 is obtained for the case where a spherical body is immersed in an infinite fluid medium and for vanishingly small total molar flux N. For finite slip... 

The total molar flux may safely be assumed to be zero so the diffusion and molar fluxes are equal. 

Toor (1957) derived a solution of the Maxwell-Stefan equations for ternary systems when the total molar flux is zero, = 0. Write down expressions for [P], (y), and show that, for = 0, the eigenvalue solutions are equivalent to the expressions given by Toor. 

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