Diffusion Stefan flow

The other mechanism appears in scrubbers. When water vapor diffuses from a gas stream to a cold surface and condenses, there is a net hydrodynamic flow of the noncondensable gas directed toward the surface. This flow, termed the Stefan flow, carries aerosol particles to the condensing surface (Goldsmith and May, in Davies, Aero.sol Science, Academic, New York, 1966) and can substantially improve the performance of a scrubber. However, there is a corresponding Stefan flow directed away from a surface at which water is evaporating, and this will tend to repel aerosol particles from the surface. 

The term p/(p-p ) derives from assumption (4.313) representing the effects of Stefan flow. If e = 1, Eq. (4.318) gives the diffusion flow in a free space. 

Rapid evaporation introduces complications, for the heat and mass transfer processes are then coupled. The heat of vaporization must be supplied by conduction heat transfer from the gas and liquid phases, chiefly from the gas phase. Furthermore, convective flow associated with vapor transport from the surface, Stefan flow, occurs, and thermal diffusion and the thermal energy of the diffusing species must be taken into account. Wagner 1982) reviewed the theory and principles involved, and a higher-order quasisteady-state analysis leads to the following energy balance between the net heat transferred from the gas phase and the latent heat transferred by the diffusing species ... 

These two conditions (Eqs. (4.97) and (4.98)) are usually sufficient for assuming the medium as quiescent in dilute systems in which both cua.s and cda,oo are small. However, in nondilute or concentrated systems the mass transfer process can give rise to a convection normal to the surface, which is known as the Stefan flow [Taylor and Krishna, 1993]. Consider a chemical species A which is transferred from the solid surface to the bulk with a mass concentration cua.oo- When the surface concentration coa,s is high, and the carrier gas B does not penetrate the surface, then there must be a diffusion-induced Stefan convective outflux, which counterbalances the Fickian influx of species B. In such situations the additional condition for neglecting convection in mass transport systems is [Rosner, 1986]... 

Diffusion of Vapor through a Stationary Gas.-Stefan Flow... 

This relation is known as Stefan s law, and the induced convective flow described that enhance.s mass diffusion is called the Stefan flow. Noting that y, — P/P and C = P/RJT for an ideal gas mixture, the evaporation rate of spccie /4 can also be expressed a.s... 

The difference in the flow rate as calculated by Eqs. (1-149) and (1-152), is the factor c/(c — c,), which is due to the additional superseding one-directional diffusion ( Stefan flux ). The amount of mass flux transferred by diffusion is therefore larger... 

Note that diffusion may cause convection Stefan flow). Figure 3.2.24 shows an example to illustrate this for the case of evaporation. Further quantitative details are given in Topic 3.2.5. 

The model was validated on NaCl solutions. In Eq. 6.3, Di is the diffusion coefficient of the solute in the liquid phase, Dg is the diffusion coefficient of the solvent vapor in the gas phase, Qi and pg are the liquid and gas densities, respectively. Sh is the dimensionless mass transfer rate in the vapor phase. This modified Sherwood number, that accounts for the film thinning effect of Stefan flow, lies typically in the range 2 < Sh < 5. The quantity Bm is the Spalding transfer number according to Abramzonand Sirignano (1989) (Sirignano (1999), compare with Eq. 1.66 in Volume... 

The theory on the level of the electrode and on the electrochemical cell is sufficiently advanced [4-7]. In this connection, it is necessary to mention the works of J.Newman and R.White s group [8-12], In the majority of publications, the macroscopical approach is used. The authors take into account the transport process and material balance within the system in a proper way. The analysis of the flows in the porous matrix or in the cell takes generally into consideration the diffusion, migration and convection processes. While computing transport processes in the concentrated electrolytes the Stefan-Maxwell equations are used. To calculate electron transfer in a solid phase the Ohm s law in its differential form is used. The electrochemical transformations within the electrodes are described by the Batler-Volmer equation. The internal surface of the electrode, where electrochemical process runs, is frequently presented as a certain function of the porosity or as a certain state of the reagents transformation. To describe this function, various modeling or empirical equations are offered, and they... 

Divisek et al. presented a similar two-phase, two-dimensional model of DMFC. Two-phase flow and capillary effects in backing layers were considered using a quantitatively different but qualitatively similar function of capillary pressure vs liquid saturation. In practice, this capillary pressure function must be experimentally obtained for realistic DMFC backing materials in a methanol solution. Note that methanol in the anode solution significantly alters the interfacial tension characteristics. In addition, Divisek et al. developed detailed, multistep reaction models for both ORR and methanol oxidation as well as used the Stefan—Maxwell formulation for gas diffusion. Murgia et al. described a one-dimensional, two-phase, multicomponent steady-state model based on phenomenological transport equations for the catalyst layer, diffusion layer, and polymer membrane for a liquid-feed DMFC. 

In the formulation and solution of conservation equations, we tend to prefer the direct evaluation of the diffusion velocities as discussed in the previous section. However, it is worthwhile to note that the Stefan-Maxwell equations provide a viable alternative. At each point in a flow field one could solve the system of equations (Eq. 3.105) to determine the diffusion-velocity vector. Solution of this linear system is equivalent to determining the ordinary multicomponent diffusion coefficients, which, in this formulation, do not need to be evaluated. 

These constraints must be satisfied in the solution of the Stefan-Maxwell equations. At a point within a chemically reacting flow simulation, the usual situation is that the diffusion velocities must be evaluated in terms of the diffusion coefficients and the local concentration, temperature, and pressure fields. One straightforward approach is to solve only K — 1 of Eqs. 3.105, with the X4h equation being replaced with a statement of the constraint. For... 

In this example, the equilibrium concentrations maintained at the interface are functions of the interface temperature, which in turn is a function of time. In addition, the velocity of the interface, v (i.e., rate of solidification), depends simultaneously upon the mass diffusion rates and the rates of heat conduction in the two phases, as may be seen by examining the two Stefan conditions that apply at the interface. For the mass flow the condition is... 

Figure 4.24 shows the reactive arheotrope trajectories according to Eq. (83) for various amounts of the liquid phase mass transfer resistance - that is, for various values of Kiiq and a low sweep gas flow rate G (at large NTt/ -values). As a result, the reactive arheotropic composition X, 02 is shifted to larger values as the liquid phase mass transfer resistance becomes more important - that is, as the value of Kuq decreases. Note that the interface liquid concentrations are in equilibrium with the vapor phase bulk concentrations. Therefore, gas phase mass transfer resistances cannot have any influence on the position of the reactive arheotrope compositions. On the other hand, liquid phase mass transfer resistances do have an effect, though the value of all binary hiq have been set equal. Again, this effect results from the competition between the diffusion fluxes and the Stefan flux in the liquid phase. 

Tables 2.12-2.14 show some values of diffusion coefficients in solids and polymers. In a flow of dilute solution of polymers, the diffusivity tensor is anisotropic and depends on the velocity gradient. The Maxwell-Stefan equation may predict the diffusion in multicomponent mixtures of polymers.

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