Interfacial fluxes

Here, J (a, t) is the instantaneous interfacial flux expressed as a state vector whose components are the diffusional and heat fluxes. Therefore, the expected value < J(a)> takes into account the variations in residence time among the entire bubble population. 

Substituting the expression for the expected interfacial flux (290) in (292), one finally obtains the required general solution ... 

The technique offers a known interfacial area under convective flow conditions that are quite well-defined, with mass transport rates that are enhanced compared to the Lewis cell and its analogs. However, in common with many other approaches, interfacial fluxes must be determined indirectly from bulk solution measurements. 

A modification of the RDC design, based on the ring-disk arrangement of the RDE [36], incorporated an arc electrode [37,38] deposited on the surface of the membrane around the untreated area. This facilitated the electrochemical detection of species reacting at the interface at short times following the reaction. This method was used to study the solvent extraction of cupric ions, which were detected by reduction to copper metal at the arc electrode. The resulting current flow was related to the interfacial flux at the membrane. 

In the majority of methods described thus far, the interfacial kinetics are deduced by measuring concentration changes in the bulk of the solution rather than at the interface, where the reaction occurs. This introduces a time lag, limiting the resolution of the measurement in the determination of interfacial kinetics. A more direct approach is to identify the interfacial flux. This can be achieved in the electrolyte dropping electrode, via the current flow, but this method is only applicable to net charge-transfer processes at externally polarized interfaces. 

Of the methodologies considered, the Lewis cell, employs direct contact of the two liquids, but does not have well-defined hydrodynamics. The constant interface cell with laminar flow has better-defined hydrodynamics, but the interfacial flux is not measured... 

The development of hydrodynamic techniques which allow the direct measurement of interfacial fluxes and interfacial concentrations is likely to be a key trend of future work in this area. Suitable detectors for local interfacial or near-interfacial measurements include spectroscopic probes, such as total internal reflection fluorometry [88-90], surface second-harmonic generation [91], probe beam deflection [92], and spatially resolved UV-visible absorption spectroscopy [93]. Additionally, building on the ideas in MEMED, submicrometer or nanometer scale electrodes may prove to be relatively noninvasive probes of interfacial concentrations in other hydrodynamic systems. The construction and application of electrodes of this size is now becoming more widespread and general [94-96]. 

Additives promote filling by diffusing to the metal surface, where they adsorb and influence the kinetics of ion-discharge and crystal-growth. The diffusion parameter can be written for an additive by replacing the current density with an interfacial flux Na-... 

The factors 4 and 4 accormt for the heterogeneity of the interface. The interfacial flux conditions. Equations (6.56) and (6.57), can be straightforwardly applied at plain interfaces of the PEM with adjacent homogeneous phases of water (either vapor or liquid). However, in PEFCs with ionomer-impregnated catalyst layers, the ionomer interfaces with vapor and liquid water are randomly dispersed inside the porous composite media. This leads to a highly distributed heterogeneous interface. An attempt to incorporate vaporization exchange into models of catalyst layer operation has been made and will be described in Section 6.9.4. 

Application of Eq. (15.1) to the liquid membrane process highlights one of the main advantages of the process, i.e., the high solute distribution coefficient that can be obtained between phases 3 and 1. However, another factor that must be considered when evaluating a separation process performance is the kinetics of transfer, which is given in a general form by Eq. (15.4). This equation indicates that the transfer rate in the contactor increases with both the interfacial flux and the specific interfacial area. 

As noted previously, most environmental flows are turbulent. The diffusive sublayer, where only diffusion acts to transport mass and the concentration profile is linear, is typically between 10 /xm and 1 mm thick. Measurements within this sublayer are not usually feasible. Thus, the interfacial flux is typically expresses as a bulk transfer... 

The CPC present in the aqueous phase is distributed between the aqueous and organic phases at the SLM-liquid interface. By maintaining low Cl ion concentration in the feed phase and high Cl ion concentration in the stripping phase, the distribution ratio of CPC (P ion form) at the aqueous feed-SLM interface can be made much higher than that at the aqueous strip-SLM interface. Under this condition, the steady-sate overall CPC flux across the membrane can be obtained from Pick s distribution law applied to aqueous diffusion film as well as the membrane itself and from interfacial reaction kinetics which describe the interfacial flux. 

In rate-based multistage separation models, separate balance equations are written for each distinct phase, and mass and heat transfer resistances are considered according to the two-film theory with explicit calculation of interfacial fluxes and film discretization for non-homogeneous film layer. The film model equations are combined with relevant diffusion and reaction kinetics and account for the specific features of electrolyte solution chemistry, electrolyte thermodynamics, and electroneutrality in the liquid phase. 

This case of the semi-infinite slab can be solved to yield both a concentration profile and an interfacial flux which ... 

Huettel, M., Ziebis, W., Forster, S., and Luther, G. W. III. (1998). Advective transport affecting metal and nutrient distributions and interfacial fluxes in permeable sediments. Geochim. Cosmochim acta. 62, 613-631. 

In the packed-bed reactor, the molar concentrations and temperature at the exterior of the catalyst particle are coupled to the respective fluid-phase concentrations and temperature through the interfacial fluxes given in Eqs. (3.3-1) and (3.3-2). Overall mass and heat transfer coefficients are often used to describe these interfacial fluxes, similar in structure to Eqs. (3.2-1) and (3.2-2). Complete solution of the packed-bed reactor model... 

If the fluxes were not equal, we would have ammonia building up at the interface. We will denote this interfacial flux by N. ... 

If A represents the interfacial area between the two liquid phases, we can calculate the interfacial fluxes as follows ... 

Having completed the formal development of an expression for the evaluation of the interfacial fluxes we turn to the actual evaluation of the zero-flux mass transfer coefficient k for some specific models of turbulence. It is usual in such developments to define the Stanton number... 

Your objective here is to examine the influence of turbulence on the interfacial fluxes of acetone(l)-benzene(2)-helium(3) for interphase mass transfer in a wetted wall column at the top of the column where the incoming vapor-gas mixture first comes into contact with the downflowing liquid mixture of acetone and benzene. The details of the column and the operating conditions are given in Example 11.5.3. For the purposes of this exercise you may ignore thermal effects. Take the vapor-phase compositions, expressed in mass fractions, in the bulk vapor and at the interface to be... 

The surface energy sources are expressed as the product of the interfacial area concentration and the averaged interfacial fluxes ... 

It is noted that two different sign conventions are in common use considering the definitions of the interfacial terms. We follow the sign convention by [54, 67], but a different set of consistent definitions are proposed in the latest reports [136, 55, 56, 57, 58]. The reason for switching the sign convention becomes clear shortly, as all the interfacial fluxes in our derivation apparently have opposite sign compared to the conventional formulations presented in this chapter. 

It is still assumed that the mean interfacial flux is proportional to the difference between the interfacial average and the intrinsic volume average of the mass fraction. In addition, the interfacial surface averaged concentrations are assumed to be in local instantaneous thermodynamic equilibrium. 

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