Linear interfaces

The usual situation, true for the first three cases, is that in which the reactant and product solids are mutually insoluble. Langmuir [146] pointed out that such reactions undoubtedly occur at the linear interface between the two solid phases. The rate of reaction will thus be small when either solid phase is practically absent. Moreover, since both forward and reverse rates will depend on the amount of this common solid-solid interface, its extent cancels out at equilibrium, in harmony with the thermodynamic conclusion that for the reactions such as Eqs. VII-24 to VII-27 the equilibrium constant is given simply by the gas pressure and does not involve the amounts of the two solid phases. 

Qualitative examples abound. Perfect crystals of sodium carbonate, sulfate, or phosphate may be kept for years without efflorescing, although if scratched, they begin to do so immediately. Too strongly heated or burned lime or plaster of Paris takes up the first traces of water only with difficulty. Reactions of this type tend to be autocat-alytic. The initial rate is slow, due to the absence of the necessary linear interface, but the rate accelerates as more and more product is formed. See Refs. 147-153 for other examples. Ruckenstein [154] has discussed a kinetic model based on nucleation theory. There is certainly evidence that patches of product may be present, as in the oxidation of Mo(lOO) surfaces [155], and that surface defects are important [156]. There may be catalysis thus reaction VII-27 is catalyzed by water vapor [157]. A topotactic reaction is one where the product or products retain the external crystalline shape of the reactant crystal [158]. More often, however, there is a complicated morphology with pitting, cracking, and pore formation, as with calcium carbonate [159]. 

Linear, one-dimensional interfaces. Perhaps the most interesting theory of promoter action is that the linear boundary on the surface between two surfaces of different composition is the seat of the catalysis, i.e. the active patch. There is abundant evidence that such linear interfaces often possess unusual reactive powers. They are undoubtedly... 

An early instance where a linear interface was found of importance for catalysis is recorded by Antropoff 2 the decomposition of hydrogen peroxide by mercury, a reaction which shows periodic variations of rate, occurring at a speed proportional to the length of the boundary of a visible skin of oxide on the mercury, and takes place at this boundary. 

Schwab and Ketsch4 have developed a theory of the kinetics of contact catalysis based on the assumption that the active patches are located at such interfaces, using the term adlineation for adsorption there but though their assumption seems a very probable one, at any rate for many cases of promoter action, it is scarcely possible to test it by comparison of rates of reaction with a theoretically derived equation there are too many adjustable constants, and other factors. One of these is the rate of surface diffusion to the active linear interfaces, or other active patches, which as many writers have pointed out, may be very considerable.5... 

Metal films condensed in a high vacuum have, usually, a much lower catalytic activity than films condensed in a low pressure of an inert gas.3 The films condensed in the gas had an oriented structure, with the (110) face parallel to the backing in the case of nickel, the (111) face in the case of iron. There was little, if any, definite orientation in the films condensed in vacuo. It has been suggested that the increased catalytic activity is, in part at least, due to the special spacing on the exposed crystal faces but there may be other reasons, perhaps the presence of linear interfaces between the bare metal and the parts covered by adsorbed gas, or a larger real surface. The suggestion is interesting, however, and now that it is possible to make faces in selected crystal planes, to the practical exclusion of others, the study of the catalytic activity of special crystal faces does not seem beyond the bounds of possibility. 

Now we shall consider the case when one interface separates two simple cubic 2D crystals composed of different molecules of types a and b interacting across the interface. In two dimensions with a linear interface (the 3D generalization is straightforward) the equations of motion for the amplitudes Amn and Bmn read (compare with eqn 9.40)... 

Interface Capturing Schemes for Free-Surface Flows, Fig. 3 Piecewise linear interface calculation (PLIC) in two dimensions... 

The formation of a microemulsion from an array of microtubes has been studied experimentally and computationally by Kobayashi and co-workers [92,93). They used the commercial code CFD-ACE, which uses a piecewise linear interface construction (PLIC) method to determine the interface. They used quarter symmetry, as the channels were eDiptical in shape, but the simulations still required 7-14 days on a 2.5 GHz Pentium IV processor. The simulations captured the main features observed experimentally, including the change of regime from continuous outflow of oil if the channel was below a critical aspect ratio, to a stream of droplets above this threshold. The model was also used to predict the droplet size as a function of oil properties and generally agreed well with the experimental data. 

The regulator specification sheet starts with a general description of the part. Here the designer can quickly determine the applicability of the part to a particular need. Looking, for example, at the Motorola linear/interface ICs databook the following information in the general description of the LM317 can be obtained ... 

Source Motorola. Linear Interface ICs Device Data, Vol. 1, 1993, pp. 3-609. 

Databooks provide the best source of information regarding voltage regulators and their use. The Motorola Linear Interface ICs Device Data VoL 1 Addendum to Sec. 10.3.3, Power Supply Circuits is an excellent source of background information. 

Motorola. 1993. Linear/Interface IC Device Databook, Vol. 1, Sec. 3 Addendum. Motorola, Inc. 

The evolution of the interface and advection of the phase-indicator function is accomplished by reconstructing the interface within each computational cell and computing the volume flux that occurs from each cell to its immediate neighbors under the prevaihng flow. The surface reconstruction problem [11] is one of finding an interface with the correct unit normal vector which divides the computational cell into two regions, each occupied by the respective fluid phase. One popular way to accomplish this is using the Piecewise Linear Interface Calculation or Con-... 

Piecewise Linear Interface Calculation (PLIC) in two dimensions. 

Figure 15.1 shows a schema comparing the real fluid configuration, where Figure 15.1c shows the simple line interface calculation (SLIC) VOF and Figure 15.Id the piecewise linear interface calculation (PLIC) VOF (the latter is also referred to as Youngs method). From these data it can be seen that, in the SLIC VOF method, the interfaces are implemented as segments parallel to the x or y axis. Although the SLIC method is advantageous in terms of simplicity and...

Figure 1S.1 Interface reconstructions of fluid shown by (a) actual configuration (b) volume fraction (c) representation with simple line interface calculation (SLIC) and (d) piecewise linear interface calculation (PLIC) [12].

Combination of the volumetric phase fraction of the dispersed phase with the interface normal then allows a Piecewise Linear Interface Reconstruction (PLIC) [24]. It allows a reconstruction of the phase geometry inside the computational cells... 

For an accurate calculation of the volume flux, the location of the interface is reconstructed with the piecewise linear interface calculation (PLIC) method from the/held [24]. 

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