Diffusion/reaction, flat interface

The maintenance of product formation, after loss of direct contact between reactants by the interposition of a layer of product, requires the mobility of at least one component and rates are often controlled by diffusion of one or more reactant across the barrier constituted by the product layer. Reaction rates of such processes are characteristically strongly deceleratory since nucleation is effectively instantaneous and the rate of product formation is determined by bulk diffusion from one interface to another across a product zone of progressively increasing thickness. Rate measurements can be simplified by preparation of the reactant in a controlled geometric shape, such as pressing together flat discs at a common planar surface that then constitutes the initial reaction interface. Control by diffusion in one dimension results in obedience to the... 

Figure 1-11 Concentration profile for (a) crystal growth controlled by interface reaction (the concentration profile is flat and does not change with time), (b) diffusive crystal growth with t2 = 4fi and = 4t2 (the profile is an error function and propagates according to (c) convective crystal growth (the profile is an exponential function and does not change with time), and (d) crystal growth controlled by both interface reaction and diffusion (both the interface concentration and the length of the profile vary).

A reaction occurring in a bulk phase will show an increase in the rate with the area as shown in Fig. 5.3 for a reaction occurring in the film or at the interface, the rate will be linearly dependent on the interfacial area. The interfacial area in a dispersed two-phase liquid-liquid system can be estimated by measuring the rate of a suitable test reaction in a reactor with the known interfacial area (a flat interface, Section 5.3.2.1), and comparing it with the reaction rate in a dispersed system [6, 15]. A convenient reactive system for this purpose is a formate ester and 1-2 M aqueous NaOH. Formate esters are very reactive to hydroxide ion (fo typically around 25 M 1 s 1), so the reaction is complete inside the diffusion film, and the reaction rate is proportional to the interfacial area. A plot of the interfacial area per unit volume against the agitator speed obtained in this way in the author s laboratory for the equipment shown in Fig. 5.12 is shown in Fig. 5.14 [8]. 

Now, it is necessary to discuss the mass transfer coefficient for component j in the boundary layer on the vapor side of the gas-liquid interface, fc ,gas, with units of mol/(area-time). The final expression for gas is based on results from the steady-state film theory of interphase mass transfer across a flat interface. The only mass transfer mechanism accounted for in this extremely simple derivation is one-dimensional diffusion perpendicular to the gas-liquid interface. There is essentially no chemical reaction in the gas-phase boundary layer, and convection normal to the interface is neglected. This problem corresponds to a Sherwood number (i.e., Sh) of 1 or 2, depending on characteristic length scale that is used to define Sh. Remember that the Sherwood number is a dimensionless mass transfer coefficient for interphase transport. In other words, Sh is a ratio of the actual mass transfer coefficient divided by the simplest mass transfer coefficient when the only important mass transfer mechanism is one-dimensional diffusion normal to the interface. For each component j in the gas mixture. 

After polarization to more anodic potentials than E the subsequent polymeric oxidation is not yet controlled by the conformational relaxa-tion-nucleation, and a uniform and flat oxidation front, under diffusion control, advances from the polymer/solution interface to the polymer/metal interface by polarization at potentials more anodic than o-A polarization to any more cathodic potential than Es promotes a closing and compaction of the polymeric structure in such a magnitude that extra energy is now required to open the structure (AHe is the energy needed to relax 1 mol of segments), before the oxidation can be completed by penetration of counter-ions from the solution the electrochemical reaction starts under conformational relaxation control. So AHC is the energy required to compact 1 mol of the polymeric structure by cathodic polarization. Taking... 

A Schottky diode is always operated under depletion conditions flat-band condition would involve giant currents. A Schottky diode, therefore, models the silicon electrolyte interface only accurately as long as the charge transfer is limited by the electrode. If the charge transfer becomes reaction-limited or diffusion-limited, the electrode may as well be under accumulation or inversion. The solid-state equivalent would now be a metal-insulator-semiconductor (MIS) structure. However, the I-V characteristic of a real silicon-electrolyte interface may exhibit features unlike any solid-state device, as... 

If the growth rate is controlled by both interface reaction and diffusion (Figure 1-1 Id), then (i) the concentration profile is not flat, (ii) the interface concentration changes with time toward the saturation concentration, (iii) the diffusion profile propagates into the melt, and (iv) the growth rate is not constant, nor does it obey the parabolic growth law. 

The mechanism of the competitive pertraction system (CPS) is presented schematically in Fig. 5.4 together with the compartmental model necessary for constructing the reaction-diffusion network. The simple flat-layered bulk liquid membrane of the thickness En and interface area S separates the two reservoirs (f, feed and s, stripping) containing transported divalent cations A2+ and B2+ (most frequently Zn2+ and Cu2+ or Ca2+ and Mg2+) and/or antiported univalent cations H+. At any time of pertraction t, their concentrations are [A]f, [B]f, and [H]f and [A]s, [Bj, and [H]s, for the feed and stripping solution, respectively. The hydrophobic liquid membrane contains a carrier of total concentration [C]. Its main property is the ability to react reversibly with cations at respective reaction zone and to diffuse throughout the liquid membrane phase. 

Zaspalis conducted a theoretical study of an asymmetric membrane with a thin, small-pore toplayer on a large-pore support, in both flat and tubular geometries for the simple isothermal reaction A - B. The best conversion was in the case where all of the catalyst was near the outer surface of the toplayer, on the reactant side. In fact, for this reaction, reactant loss meant that the membrane reactor did worse than the fixed bed reactor. He also claimed that the optimal distribution is a delta function, for both geometries, for all possible kinetics, when diffusion of a reactant occurs from one side only. For segregated reactants, for the reaction A + B - C as illustrated in Figure 22, the optimal location of the catalyst was at the toplayer/support interface, assuming that diffusion was the only transport mechanism through the support. 

We consider the situation depicted in Figure 4.5, involving transport through a gas film to a liquid interface, followed by diffusion and reaction in the liquid film. The difference element over which the mass balance is taken is the same as that for the flat-plate catalyst pellet, and leads to the same differential equation and the same boxmdary conditions ... 

Electrodes are usually composed of a gas diffusion layer with thin catalyst coatings at the electrode-electrolyte interfaces. Hydrogen and oxygen reactants are supplied to the anode and cathode electrodes surfaces. The original rod-type electrodes are generally replaced with flat or circular annular surfaces to increase the contact surface area for reactions. The structure of the electrode is... 

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