Propagating fronts

If the diffusion coefficient of species A is less tlian tliat of B (D < D ) tlie propagating front will be planar. However, if is sufficiently greater than tire planar front will become unstable to transverse perturbations and chaotic front motion will ensue. To understand tire origin of tire mechanism of tire planar front destabilization consider tire following suppose tire interface is slightly non-planar. We would like to know if tire dynamics will tend to eliminate this non-planarity or accentuate it. LetZ)g The situation is depicted schematically in figure... 

This reaction undergoes conversion in one sequence of consecutive elementary reaction steps and so only one propagating front is formed in a spatially distributed system [68]. Depending on the initial ratio of reactants, iodine as colored and iodide as uncolored product, or both, are formed [145]. 

We will now estimate the front velocity using the method of Guy (1984). Let us consider, in one dimension, the conservation of element i in a rock column (x = xt to x = x2) of unit section which contains a propagating front at the time-dependent position s(t). The discontinuity is handled by breaking the column at s = s(t). The amount of element i in the rock column only changes by fluid exchange through both ends of the column, hence... 

For a mutation that occurs at the very edge of the total population wave, 0, the transport speed is equal to the speed of propagation front, v = c the two waves, of the total population and of the mutation, are synchronized. 

From Eqs. (51) and (53) it follows that = 2, a value in good agreement with the numerical simulations of Edmonds, Lillie, and Cavalli-Sforza [22], which lead to ( = 2.2. The difference of 0.2 between theory and simulations is due to the random drift, which was taken into account in the simulations but is neglected in our theory. By including the random drift, our theory provides information about the details of the motion of the propagation front [23]. 

The conversion process occurs both on macro- and micro-scale, that is, on single particle level and on bed level. In other words, the conversion process has both a macroscopic and microscopic propagation front. One example of the macroscopic process structure is shown in Figure 10. The conversion front is defined by the process front closest to the preheat zone, whereas the ignition front is synonymous with the char combustion front. 

Propagating Fronts of Polymerization in the Physical Chemistry Laboratory 125... 

I. Rubinstein, Asymptotics of propagating front formation in diffusion kinetics, SIAM J. Appl. Math., 45 (1985), pp. 403-419. 

Hanna, A., Saul, A., and Showalter, K. (1982). Detailed studies of propagating fronts in the iodate oxidation of arsenous acid. J. Am. Chem. Soc., 104, 3838-44. 

Multiplicity and Propagating Fronts in Adiabatic and Nonadiabatic Fixed-Bed Reactors... 

For a numerical simulation the one-phase one-dimensional model has been used. The model failed to predict in nonadiabatic case multiplicity of propagating fronts and erratic behavior as well. 

The phenomenon of multiplicity and propagating fronts in adiabatic fixed bed reactors has received much attention in the literature and is the subject of a rather exhaustive treatment [1-6]. Unlike the adiabatic operation, the nonadiabatic case enjoyed far less attention and many questions are still to be answered. Hence, the principal interest in this work was to investigate experimentally the theoretically the characteristic features of multiplicity and propagating fronts created under different conditions in a nonadiabatically operated packed bed reactors and to make a comparison with the adiabatic operation. 

Investigation of the propagating fronts for nonadiabatic conditions shown that the front velocity is not constant and depends on the position of the front in the reactor [15]. For a downstream propagating front, the velocity, hot spot temperature and exit conversion exhibited an oscillatory character [7]. 

Kinetic transitions between well-defined micromorphologies are usually dominated not by second order but by first order thermodynamics. Recent ideas have shown how propagating fronts of the new nucleating phase may be responsible for the limiting rate. ... 

As a rule, an increase in temperature in the course of polymerization is accompanied, by various kinetic effects. For example, in the radical polymerization of vinyl monomers changes can take place in the concentration of radicals and the time when the gel effect sets in. In addition a process of degradation can be superimposed on the polymerization process. The temperature and conversion non-uniformities occurring in the course of polymerization can change the thermal process itself, converting bulk polymerization into a reaction with propagating front, and vice versa. 

Fig. 33 A series of nanosecond time-resolved interferometric images of the triazene polymer film at a fluence of 250 mj cm 2, which were obtained with the surface configuration. In this case, fringe shift to the right represents an etching. Delay time (At) (a) -9 ns, (b) +10 ns, (c) +57 ns, (d) +260 ns, (e) +l s, and (f) +3 /is. The black bar in (f) indicates 1 mm in the image. Arrows in (e) and (f) point to the propagation front of the shock wave. REPRINTED WITH PERMISSION OF [Ref. 125], COPYRIGHT (1997) American Chemical Society...

An order-of-magnitude estimate of the critical stirring rate can be obtained from the balance between the typical flow velocity and the speed of the propagating fronts. The assumption behind this is that the flow produces a topological transition by generating a quasi-one-dimensional filamental structure. This naturally leads to the break down of coexistence, due to the different invasion velocities of the fronts separating the patches of different strains. 

In the experiment shown in Fig. 8.8 the propagating fronts all have their origin at the rim of the unsealed surface. Thus, also crevice corrosion beneath the insulating lacquer and the associated acidification of the chemical environment could have affected our results [24], To exclude this possibility the stainless steel sample was taken out of the electrolyte and imaged ex situ after termination of the experiment and careful removal of the insulating lacquer (see Fig. 8.9). The pits identified by arrows and numbers 1, 2, and 3 are the initial pits which released enough aggressive ions to weaken the oxide layer. It is possible that the first pits nucleated near the... 

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