Solid molar flux equations

We finally decided to solve the problem by using a simple sifting procedure similar to the one used by Emanuel and Vale (18). Before the integrator was called, the derivatives were determined for the solids stream molar flux equations. Any flux equation that had a derivative below the value of the sift parameter was considered to be inactive at that time. The energy flux equation was always considered active. Only the active equations... 

In the DGM, the solid phase is modeled as giant dust molecules held motionless in space with which the diffusing gas molecules collide. The constitutive equations governing the diffusion molar flux intensities Nf for both MTPM and DGM are the generalized Maxwell-Stefan equations... 

Different combinations of spatial finite difference formulas were tried to determine the best set for our system of equations. The two point upwind formula was found to be best for the solids component molar fluxes. The low order formula was used because most of the gasifier reactions turn off abruptly when a component disappears and this creates sharp discontinuities. Higher order formulas tend to flatten out discontinuities, and in some cases, this causes material balances to be lost which then leads to numerical instability problems. Maintaining component material balance is an important aid to preserving numerical stability in the calculations. The low order formulas minimized these difficulties. 

We now proceed to perform our shell balance on A. The area that appears in the balance equation is the total area (voids and solids) normal to the direction of the molar flux ... 

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. 

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. 

Equation (2.2.1) implies that the molar flux is proportional to the concentration difference, which is a reasonable approximation for many gas-solid reaction systems. A somewhat more accurate mass transfer equation that has been proposed by Bird et al. [1, Chapter 21] takes the form... 

In equation 6.85, ([J (x, r)] nL) is the total liquid-solid particles interface, averaged, molar diffusive flux of component f. Note that Ag - = Asr +Asjf The second term of the right-hand side of equation 6.85 is 0 because at the catalyst interface convective fluxes are 0. On the other hand, from equation 6.13 the molar diffusive flux through the boundary layer at the liquid-solid interface must be equal to the reaction rate at the liquid-solid interface ... 

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