Molar flux expressions

Looking back to our molar flux expression (Equation 3.43),... 

The definitions of the mass, molar, and volume average bulk velocities are given in Table 1 along with selected mass and molar flux expressions related to each of the specified reference velocities for a binary system of components A and B. The mutual diffusion coefficient Dab is the diffusivity of component A and B in the mixture as defined below. The mutual diffusion coefficients appearing in Table 1 are identical in all of the expressions and Dab = DBA-... 

Figure 3.1.3B. Explanation for molar flux expression (3.1.77) of species L Replace Up, in Figure 3.1.3A by v, particles by molecules of species i and particle number density N, by Q, ifee molar density of i in the volume element ABCDA Bf OF/. We obtain Nfe = CiViz, if we follow the explanation provided in the caption to Figure 3.1.3A...

Diffusivity measures the tendency for a concentration gradient to dissipate to form a molar flux. The proportionality constant between the flux and the potential is called the diffusivity and is expressed in m /s. If a binary mixture of components A and B is considered, the molar flux of component A with respect to a reference plane through which the exchange is equimolar, is expressed as a function of the diffusivity and of the concentration gradient with respect to aji axis Ox perpendicular to the reference plane by the fpllqvving relatipn 6 /... 

The first equahty (on the left-hand side) corresponds to the molar flux with respect to the volume average velocity while the equahty in the center represents the molar flux with respect to the molar average velocity and the one on the right is the mass flux with respect to the mass average velocity These must be used with consistent flux expressions for fixed coordinates and for Nc components, such as ... 

A solute diffuses from a liquid surface at which its molar concentration is C, into a liquid with which it reads. The mass transfer rate is given by Fick s law and the reaction is first order with respect to the solute, fn a steady-state process the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration C of solute at a depth z from the surface in terms of the molecular diffusivity D and the reaction rate constant k. What is the molar flux at the surface ... 

As discussed earlier, the mean drift velocity is the volume flux, Jv [see Eq. (70)]. Using the ideal gas equation to relate volume flow to molar flow [see Eq. (71)], the relationship between mean drift velocity and molar flux J may be written as um = (RTIPa)J. With this expression for um, Eqs. (80) and (81) are combined to give the desired expression for molar flux,... 

It is often convenient to express the molar flux for mass transfer in terms of a mass transfer coefficient, and in these circumstances Eq. (10) can be written as... 

If the ideas are used that an infinite sea of liquid surrounds each gas bubble and a pseudo-steady state exists, then the molar flux can be described by the familiar expression... 

Some aspects of the theoretical development which have been presented here follow along the lines of an important paper by Newman and Simon (1980). Their analysis differs from the simplified analysis presented here in two respects. First, the idea that ttd, is a measure of the type of bubble growth which occurs was not incorporated in the Newman-Simon analysis. Second, Newman and Simon used a more realistic expression for the molar flux. ... 

Reid, Sherwood and Prausnitz [11] provide a wide variety of models for calculation of molecular diffusion. Dr is the Knudsen diffusion coefficient. It has been given in several articles as 9700r(T/MW). Once we have both diffusion coefficients we can obtain an expression for the macro-pore diffusion coefficient 1/D = 1/Dk -i-1/Dm- We next obtain the pore diffusivity by inclusion of the tortuosity Dp = D/t, and finally the local molar flux J in the macro-pores is described by the famiUar relationship J = —e D dcjdz. Thus flux in the macro-pores of the adsorbent product is related to the term CpD/r. This last quantity may be thought of as the effective macro-pore diffusivity. The resistance to mass transfer that develops due to macropore diffusion has a length dependence of R]. 

For mass transfer in the gas phase, the molar flux of a particular component N (in kmolm s ) is related to the concentration difference in the gas phase AC, expressed in terms of molar concentration (kmol m ), by... 

In Equation (2.60) the membrane flux,. /, is a mass flux (g/cm2 s), whereas the gas separation literature predominantly uses a molar flux, typically expressed in the units cm3(STP)/cm2 s. The molar flux, j, can be linked to the mass flux, Ji, by the expression... 

For a binary mixture, the convective molar flux, Jc, for species A can be expressed as... 

For multicomponent mixtures, the overall molar flux for species i in the mixture can be expressed by... 

Neglecting convection effects, the solution-diffusion model gives the following expressions for water (1) and salt (2) molar fluxes through a membrane with a selective layer thickness of L and a transmembrane pressure drop Ap (Merten, 1966) ... 

The molar flux densities VI)W and N,i(. can be expressed according to Fick s first law ... 

The rate of transfer, expressed as a molar flux Na (mol m 2 s-1), is proportional to the driving force, which with the film model gives ... 

The driving force for the transport is provided by a concentration gradient as the reactant moves further towards the center of the pellet its concentration is decreased by reaction. The resistance to the transport mainly originates from collisions of the molecules, either with each other or with the pore walls. The latter dominate when the mean free path of the molecules is larger than the pore diameter. Usually both type of collisions are totally random, which amounts to saying that the transport mechanism is of the diffusion type. Hence the rate of transport, expressed as a molar flux in mol mp2 s-1, in the case of equimolar counterdiffusion can be written as ... 

When the size of the pores is much smaller than the molecular mean free paths in the gas mixture, collisions of gas molecules with the pore wall are more frequent than inter-molecular collisions. This type of gas transport is also known as Knudsen diffusion. In this case each molecule acts independently of all the others and each component in a mixture behaves as though it were present alone. The movement of each molecule can be conveniently pictured as a random walk between the walls of pores. This leads to the following expression for the molar flux of species A ... 

At an O/W interface, however, molecule B in the W phase can approach only from the W-phase side to molecule A staying in contact with the 0-phase side of the interface. It is here assumed that molecule B reacts with molecule A just when it reaches the reaction surface , i.e., the part of spherical surface of the radius of tab around the centre of A which bulges out to the W phase (see the shadow part in Figure 8.8b). The diffusion-controlled molar flux of B towards the reaction surface, (,ep would be obtained by analogy of Equation (16). However, the relative diffusion coefficient Dab in Equation (16) should be replaced by the absolute diffusion coefficient of B in the W phase (D ), because in this case, molecule A is regarded as staying at the interface for a reaction with B. Consequently, can be expressed by... 

As we have already indicated, the upper line of Equation 3.7 is a representative example from the large class of expressions relating various flows to their causative forces. In this particular case, Jj is the rate of flow of moles of species ) across unit area of a plane and can be expressed in mol m-2 s-1. Such a molar flux density of a species ) divided by its local... 

The simplest approach is to say that the molar fluxes at a vapor-liquid interface may be expressed as... 

The movement of a contaminant near the surface of a part to the bulk fluid is governed by mass transport mechanisms. The molar flux of a species A from a surface may be expressed in terms of a composition driving force and a mass transfer coefficient ... 

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... 

Differentiate this concentration profile to obtain an expression for the molar flux of A. 

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