Diffusional mass flux

In this book we limit our treatment to dilute solutions so that the diffusional mass flux is small. In this way the existence of diffusion does not appreciably alter the fluid motion, so that the velocity and stress boundary conditions can be considered to be unaltered. Treatments of diffusion with high mass fluxes appear elsewhere (B3, S3, S4). 

The diffusional mass flux to the collector from the solution of Eq. (4.3.19)... 

Hence, j, = 0, which suggests that the sum of all diffusional mass fluxes in the mixture with respect to the mass-averaged frame is zero. For binary mixtures, this statement is known as equimolar counterdijfusion. 

When there are only two components in the mixture, diffusional fluxes are written in terms of a driving force and a binary molecular diffusion coefficient via Pick s first law. For example, if the diffusional mass flux with respect to a reference frame that translates at the mass-averaged velocity of the mixture is based on a mass fraction driving force, then Pick s first law for component A is... 

The diffusional molar flux of component A is expressed via Pick s law in terms of the concentration gradient of A, only. Coupling between the diffusional mass flux of one species and all the independent mass fractions in the liquid phase is avoided by modeling this multicomponent diffusion problem as if it were a pseudobinary mixture. 

Step 9. Use matrix concepts for positive-definite quadratic forms and the Onsager reciprocal relations to develop expressions for the diffusional mass flux of component i and the molecular flux of thermal energy in binary mixtures. 

Step 10. Use the expression for diffusional mass flux in binary mixtures to derive the Stokes-Einstein diffusion equation for liquid-phase diffusivities. 

The definition of diffusional mass flux j, with respect to the mass-average velocity v of the mixture was stated in equation (25-31) as ... 

COUPLING BETWEEN DIFFUSIONAL MASS FLUX AND MOLECULAR FLUX OF THERMAL ENERGY IN BINARY MIXTURES THE ONSAGER RECIPROCAL RELATIONS... 

COMPLETE EXPRESSION FOR THE DIFFUSIONAL MASS FLUX OF COMPONENT A IN A BINARY MIXTURE... 

The linear law for diffusional mass flux in (25-60) that obeys the Curie restriction for coupling between fluxes and forces of the same tensorial rank is... 

Now, the diffusional mass flux of species A, given by equation (25-70), is written in terms of V( a, VT, and the external force field as... 

These definitions introduce the binary molecular ( ab). thermal (kj), and pressure (kp) diffusion coefficients. The final form for the diffusional mass flux of species A in a binary mixmre with respect to the mass-average velocity v is... 

This calculation precludes development of the Einstein diffusion equation for forced diffusion in the presence of a gravitational field. The coefficient of (ge — gA) in equation (25-77) for the diffusional mass flux of species A can be evaluated via thermodynamics. The extensive Gibbs free energy of a one-phase binary mixture with 3 degrees of freedom requires four independent variables for complete description of this thermodynamic state function. Hence, ( (T, p, N/, Nb) is postulated where Ni represents the mole numbers of species i, and the total differential of is... 

CONNECTION BETWEEN TRANSPORT PHENOMENA AND THERMODYNAMICS FOR DIFFUSIONAL MASS FLUXES AND DIFFUSIVITIES IN BINARY MIXTURES... 

This classic equation, which combines well-known results from mass transfer and low-Reynolds-number hydrodynamics, is very useful to predict the effect of molecular size on diffusion coefficients. The assumptions that must be invoked to arrive at the Einstein diffusion equation and the Stokes-Einstein diffusion equation are numerous. A single spherical solid particle of species A experiences forced diffusion due to gravity in an infinite medium of fluid B, which is static. Concentration, thermal, and pressure diffusion are neglected with respect to forced diffusion. Hence, the diffusional mass flux of species A with respect to the mass-average velocity v is based on the last term in equation (25-88) ... 

These considerations produce the following result for the diffusional mass flux of species A ... 

By definition of the mass-average velocity v of the mixture, all diffusional mass fluxes with respect to v must sum to zero. Hence, Ja = —jn for binary mixtures. The final expression for the molecular flux of thermal energy in binary mixtures, neglecting the diffusion-thermo (i.e., Dufour) effect, is... 

These conditions on the diffusional mass flux and the molecular flux of thermal energy, the latter of which includes Fourier s law and the interdiffusional contribution, allow one to relate temperature and reactant molar density within the pellet. If n is the local coordinate measured in the direction of n, then equations (27-20) and (27-21) can be combined as follows ... 

The final step is to express the diffusional mass flux of reactant A in terms of an effective diffusivity and the gradient of Ca via equations (27-24) and (27-25) ... 

Pick s law for diffusional mass flux j, peuet within the catalytic pores is written in terms of a concentration gradient, molecular weight, and the effective diffusivity... 

When the stoichiometric relation given by (30-5) is evaluated at the external surface of the catalyst, it is possible to invoke continuity across the gas/porous-solid interface and introduce mass transfer coefficients to evaluate interfacial fluxes. Diffusion and chemical reaction within the catalytic pores are consistent with the following stoichiometric relation between diffusional mass fluxes ... 

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