Molar flux expressions derivation

In deriving these expressions we ignored the fact that the elements of [/< ] are complicated functions of the mole fractions and of the molar fluxes. It would not be at all straightforward to allow for this dependence. In practice, it is found that the rate of convergence of Newton s method is not seriously impaired by this simplification. 

When taking these partial derivatives it must be remembered that, in general, the molar densities, the mass transfer coefficients, and thermodynamic properties are functions of temperature, pressure, and composition. In addition, H is a function of the molar fluxes. We have ignored most of these dependencies in deriving the expressions given above. The important exception is the dependence of the K values on temperature and composition that cannot be ignored. The derivatives of the K values with respect to the vapor mole fractions are zero in this case since the model used to evaluate the K values is independent of the vapor composition. 

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. 

In Section 12.2.2 we derived an expression that allows us to calculate the average molar fluxes in a vertical slice of froth on a tray under the assumptions that the vapor rises through the froth in plug flow and the liquid in the vertical slice is well mixed. Extend the treatment and derive an expression for the average mass transfer rates for the entire tray if the liquid is in plug flow. Some clues as to how to proceed may be found in Section 13.3.3. 

Equation (1-77) is one of the most fundamental relations in the analysis of mass transfer phenomena. It was derived by integration assuming that all the molar fluxes were constant, independent of position. Integration under conditions where the fluxes are not constant is also possible. Consider, for example, steady-state radial diffusion from the surface of a solid sphere into a fluid. Equation (1-70) can be applied, but the fluxes are now a function of position owing to the geometry. Most practical problems which deal with such matters, however, are concerned with diffusion under turbulent conditions, and the transfer coefficients which are then used are based upon a flux expressed in terms of some arbitrarily chosen area, such as the surface of the sphere. These matters are discussed in detail in Chapter 2. 

Fig. 5. Flux analysis data [12,31,32] on C3-C4 conversions in isogenic strains derived from lysine-producing C. glutamicum MH20-22B in chemostat cultures revealing a strong correlation with the L-lysine production rate. Rates are molar and expressed as % of the glucose uptake rate...

The expressions that we obtained for the molar flux of very slow, slow, normal, fast, and infinitely fast reactions are inserted into the mass balances of the ideal reactor models. The molar flux at the gas-liquid interface was derived for ideal reactor models for plug flow column reactors (Equations 7.15 and 7.16), for stirred tank reactors (Equations 7.22,7.25, and 7.26), and for BRs (Equations 7.33 and 7.34) ... 

The expression for molar flux, N[j, was derived for different kinds of reactions. The flux from the liquid film to the liquid bulk (required for the mass balances) is equal to the flux from the liquid film to the solid surface, Nlj, for very slow reactions (no reaction in the liquid film). For other types of reactions, the flux is obtained from the concentration profile of the liquid bulk cla(z) by calculating the derivative dcLA/dz and inserting it into Equation 7.36. 

One more algebraic eqnation is required to solve for all unknown molar densities at Zk+i It is not advantageons to write the mass balance at the catalytic surface (i.e., at xatx-i-i) because the no-slip boundary condition at the wall stipnlates that convective transport is identically zero. Hence, one relies on the radiation boundary condition to generate eqnation (23-46). Diffusional flux of reactants toward the catalytic snrface, evalnated at the surface, is written in terms of a backward difference expression for a first-derivative that is second-order correct, via equation (23-40). This is illnstrated below at Xwaii = x x+i for equispaced data ... 

Now, in a multicomponent system, the variation of the chemical potential with space can be expressed in terms of the molar fractions, or concentrations as function of space. Further the velocity of the particles can be expressed in terms of a material flux across an imaginary perpendicular surface to the respective axis. In this way, the equation of diffusion can be derived from thermodynamic arguments. We emphasize that we have now silently crossed over from equilibrium thermodynamics to irreversible thermodynamics. 

The advantages of using the special names and symbols of SI derived units are apparent in Table 4. Consider, for example, the quantity molar entropy the unit J/(mol K) is obviously more easily understood than its SI base-unit equivalent, m kg s K moP. Nevertheless, it should always be recognized that the special names and symbols exist for convenience either the form in which special names or symbols are used for certain combinations of units or the form in which they are not used is correct. For example, because of the descriptive value implicit in the compound-unit form, communication is sometimes facilitated if magnetic flux (see Table 3a) is expressed in terms of the volt second (V s) instead of the weber (Wb). 

An expression for the chemical diffusion coefficient for binary diffusion by means of vacancies can be derived in a straightforward way. Assume that the molar volume is independent of concentration. Since transport occurs via vacancies, there will be a flux of vacancies in addition to the fluxes of the components 1 and 2 in the lattice system. If the jump frequency r of particles of type 1 into the vacancies is greater than that of particles of type 2, then a local flux of vacancies will occur towards the region of higher concentration of component 1. Under the assumption of internal thermodynamic equilibrium, these vacancies are removed from the crystal at sites of repeatable growth (i.e. dislocations, grain boundaries). Because of this flux... 

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