MOLECULAR MASS TRANSFER

As early as 1815 it was observed qualitatively that whenever a gas mixture contains two or more molecular species, whose relative concentrations vary from point to point, an apparently natural process results which tends to diminish any inequalities in composition. This macroscopic transport of mass, independent of any convection effects within the system, is defined as molecular diffusion. 

Consider a hypothetical section passing normal to the concentration gradient within an isothermal, isobaric gaseous mixture containing solute and solvent molecules. The two thin, equal elements of volume above and below the section will contain the same number of molecules, as stipulated by Avogadro s law (Welty et al., 1984). 

---

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

In the internal wake region W(2), one can neglect molecular mass transfer in the radial direction. 

Any catalyzed reaction proceeds in parallel with a variety of molecular mass transfer phenomena, so that in many cases the observed activities and product distributions are linked to the diffusional behavior of the reactant, intermediate, and product molecules [1-3]. Knowledge of the intrinsic diffusivities is, therefore, particularly relevant for understanding the overall processes of molecular conversion, as well as for optimization of industrial plants. 

Reactions catalyzed by zeolites and other solid catalysts proceed in highly heterogeneous systems. The determination of local diffusivities in such systems by conventional gravimetric or flow methods [3-5] is very complicated and incurs the risk of misinterpretation, since all these methods are sensitive to the response of the whole system rather than to the mobility of individual species within well-defined regions. With reference primarily to diffusion studies in zeolitic adsorbent-adsorbate systems, the present chapter will show how NMR spectroscopy is a most versatile tool for investigating molecular mass transfer phenomena in heterogeneous catalysis. Information may be provided with respect to both microscopic and macroscopic dimensions, involving the observation of molecular distributions as well as the diffusion paths of individual molecules. 

The variety of diffusion mechanisms involved in intracrystalline molecular mass transfer is most vividly reflected in the different patterns of the concentration dependence of intracrystalline self-diffusion. A classification of the various concentration dependences so far observed by PFG NMR is presented in Fig. 10. 

In many cases the rate of molecular mass transfer through the bed of zeolite crystallites (see Sec. III.D.) is found to be so fast that the propagator determined in the PFG NMR experiment may be easily separated into its two constituents. 

This is a surface-related phenomenon based on the mass flux vector of component i and the surface area across which this flux acts. Relative to a stationary reference frame, p, v, is the mass flux vector of component i with units of mass of species i per area per time. It is extremely important to emphasize that p, v, contains contributions from convective mass transfer and molecular mass transfer. The latter process is due to diffusion. When one considers the mass of component i that crosses the surface of the control volume due to mass flux, the species velocity and the surface velocity must be considered. For example. Pi (Vr — Vsurface) is the mass flux vector of component i with respect to the surface... 

No assumptions have been invoked to obtain this result. As illustrated below, the mass flux term with respect to a stationary reference frame, V p, v, contains contributions from bulk fluid flow (i.e., convection) and molecular mass transfer via diffusion. In fact, whenever the divergence of a flux appears in a microscopic balance expression, its origin was a dot product of that flux with the outward-directed unit normal vector on the surface of the control volume, accounting for input and output due to flux across the surface that bounds V(t). The divergence of a flux actually represents a surface-related phenomenon that has been transformed into a volume integral via Gauss s law. 

Rate of reactant transport toward the catalytic surface via molecular mass transfer... 

Molecular mass transfer in equation (23-59) is integrated as follows ... 

Symmetry conditions at x = 0 and y = 0 nullify any contributions from molecular mass transfer across these boundaries. Hence,... 

Sherwood number (Sh) A dimensionless measure of the ratio of convective mass transfer to molecular mass transfer. If the mass transfer coefficient k is defined in terms of the film theory, then Sh is a measure of the ratio of hydraulic diameter to the thickness of the boundary layer. See Section 6.5. 

Prandtl number [ratio of molecular momentum transfer (friction, or viscosity effect) to molecular heat transfer (heat conduction)] Schmidt number [ratio of molecular momentum transfer (friction or viscosity effect) to molecular mass transfer (diffusion effect)]... 

Dimensionless numbers used to calculate the transfer coefficients a-l Nu = — K Nusset number (ratio of total heat transfer over heat conduction alone) P-l Sh = — D Sherwood number or second Nusset number (ratio of total mass transfer over molecular mass transfer)... 

We can treat mass transfer in a manner somewhat similar to that used in heat transfer with Fourier s law of conduction. However, an important difference is that in molecular mass transfer one or more of the components of the medium is moving. In heat transfer by conduction the medium is usually stationary and only energy in the form of heat is being transported. This introduces some differences between heat and mass transfer that will be discussed in this chapter. 

The values of some of the parameters in these equations, such as the diffusion coefficient D and the characteristic length parameter d, will depend on specific models and definitions (see below). Using the definitions of the Sherwood number, Sh = kd/D, the ratio of total and molecular mass transfer (with k the mass transfer coefficient), and the Schmidt number. Sc = r]/pD the ratio of momentum and molecular mass transfer, the equation can be written as ... 

---