Diffuse reaction zone models

The model to be presented here is based on the work of Sohn and Szekely [23-26], which may be regarded as a generalization of the numerous models [27, 28, 31, 32, 34, 35] that have been proposed to represent the diffuse reaction zone of reacting porous solids. Henceforth we shall refer to the model as the grain model. ... 

Recall that we are assuming faem "C faff (°r fax, if turbulent flow). Anyone who has carefully observed a laminar diffusion flame - preferably one with little soot, e.g. burning a small amount of alcohol, say, in a whiskey glass of Sambucca - can perceive of a thin flame (sheet) of blue incandescence from CH radicals or some yellow from heated soot in the reaction zone. As in the premixed flame (laminar deflagration), this flame is of the order of 1 mm in thickness. A quenched candle flame produced by the insertion of a metal screen would also reveal this thin yellow (soot) luminous cup-shaped sheet of flame. Although wind or turbulence would distort and convolute this flame sheet, locally its structure would be preserved provided that faem fax. As a consequence of the fast chemical kinetics time, we can idealize the flame sheet as an infinitessimal sheet. The reaction then occurs at y = yf in our one dimensional model. 

As will be shown for the CD model, early mixing models used stochastic jump processes to describe turbulent scalar mixing. However, since the mixing model is supposed to mimic molecular diffusion, which is continuous in space and time, jumping in composition space is inherently unphysical. The flame-sheet example (Norris and Pope 1991 Norris and Pope 1995) provides the best illustration of what can go wrong with non-local mixing models. For this example, a one-step reaction is described in terms of a reaction-progress variable Y and the mixture fraction p, and the reaction rate is localized near the stoichiometric point. In Fig. 6.3, the reaction zone is the box below the flame-sheet lines in the upper left-hand corner. In physical space, the points with p = 0 are initially assumed to be separated from the points with p = 1 by a thin flame sheet centered at... 

Although there is plenty of experimental evidence that ultrasound improves leaching the exact mechanism is not fully understood. Swamy and Narayana [60] have suggested models for leaching in the presence and absence of ultrasound (Fig. 4.4). Normal leaching takes place as the solvent front moves inward and a steady state diffusion occurs through the depleted outer region and is equal to the rate of reaction within the reaction zone itself (Fig. 4.4a). 

In the first scheme the metal boiling point is less than the oxide boiling point and the model consists of a vaporising droplet of metal surrounded by a detached reaction zone where condensed oxides appear as fine droplets. The reaction rate is said to be controlled by the vapour phase diffusion of metal and atmospheric oxygen into the reaction zone as in Figure 5.6. 

Part of the radiation from the reaction Zone of this flame is nonthermal, being chemiluminescent in origin. In determination of populations of species such as C2 and CH in low-pressure diffusion or premixed flames at 1-15 torr, these authors developed a simple model of a flame which reflects the characteristic requirements for start of laser action. They show that by choice of the proper experimental parameters in the model, laser action should be attainable... 

Figure 6.3a shows the idealized sketch of concentration profiles near the interface by the 1 latta model, for the case of gas absorption with a very rapid second-order reaction. The gas component A, when absorbed at the interface, diffuses to the reaction zone where it reacts with B, which is derived from the bulk of liquid by diffusion. Ihe reaction is so rapid that it is completed within a very thin reaction zone this can be regarded as a plane parallel to the interface. The reaction product diffuses to the liquid main body. The absorption of CO2 into a strong aqueous KOH solution is close to such a case. Equation 6.21 provides the enhancement... 

Fig. 9.7. Non-stationary behaviour in the diffusive autocatalysis model showing sustained temporal and spatial oscillations with D = 5.2 x 10 3, / = 0.08, and k2 = 0.05 (a) z = 0 or 465 (the oscillatory period) (b) z = 115 (c) z = 140 (d) z = 160 (e) z = 235. The limit cycle obtained by plotting the concentrations at the centre of the reaction zone, a,s(0) and /J (0), versus each other is shown in (f). The broken curve in (a) is the unstable stationary-state profile about...

We may also briefly consider the behaviour of the simple autocatalytic model of chapters 2 and 3 under reaction-diffusion conditions. In a thermodynamically closed system this model has no multiplicity of (pseudo-) stationary states. We now consider a reaction zone surrounded by a reservoir of pure precursor P. Inside the zone, the following reactions occur ... 

Murray (1982) has confirmed this pattern of behaviour empirically for a variety of two-variable models with zero-flux boundary conditions such as those considered here. In general, the dominant mode increases in wave number n as the size of the reaction zone y increases, but decreases as the ratio of diffusivities increases—as shown in Fig. 10.6. 

Combustion models which consider the thickness of the reaction zone usually accentuate cither heat conduction mechanisms (thermal theory) or the diffusion mechanisms (diffusion theory) and the models are of necessity of limited value. Simpler models in which the reaction zone or flame front is considered to be an infinitesimally thin discontinuity in the flow, while not simulating exactly the observed conditions, allow the model to be of more general utility and many combustion phenomena become easier to understand because of this simplification. It is the latter approach which is discussed first in this paper—i.e., the combustion process is regarded as a wave phenomenon. 

The control volume of the surface reaction zone, which is at the surface of the growing film (Figure 12), links the physical situation with the mathematical model that follows. Because the control volume is small enough, the incident flux from the sources is uniform within this volume. The net rate of surface diffusion into the control volume is assumed to be negligible compared with the incident flux. An incident component entering the control volume at a rate r(i, j) is either adsorbed or reflected from the surface, where the rate of reflection is r(r, j). An adsorbed component may react at a rate r(rxt, j) to form a compound, be emitted from the surface into the gas phase at a rate r(e, j), or be codeposited with the compound in an elemental form at a rate r(d, j). 

With the establishment of the primary phototriplet reduction mechanism we now turn to the explanation for the effect of flow rates and the formation of polarized phenoxy radicals. Since reaction [2] is a relatively fast secondary process it is readily understood that the observation of the primary ketyl radicals will be dependent upon flow rate. The triplet polarization (E) of the secondary phenacyl radical should not have been affected but the increased contribution of the E/A Radical-Pair polarization altered the overall appearance of the polarization pattern. The diffusion model of the Radical-Pair theory relates the E/A polarization magnitude to the radical concentration within the reaction zone. Since the phenacyl radical is considered to be very chemically reactive, and the product phenol "accumulated" within the reaction zone is also a much better hydrogen donor, the following reactions will proceed within the reaction zone ... 

Fig. 3.40(a), top]. In situation II ( AG/ = AGj > Xc) the effect is completely lost since both reaction zones have exactly the same shape [Fig. 3.40(b), top]. Thus the initial ion distribution, even when it coincides with one of them, cannot be inside the other. As a consequence, only the descending (diffusion-controlled) branch of this dependence is seen in Figure 3.40(a) (bottom). Such a high sensitivity of the results to the shape and relative location of the ionization and recombination zones makes any model simplifications of these zones undesirable. 

If reaction products are less volatile, then their condensation can influence the combustion mechanism. For example, although boron is less volatile than B2O3, this oxide is sufficiently nonvolatile for its liquid phase to play a role in the combustion of boron particles under many circumstances [54]. Relatively volatile fuels with nonvolatile combustion products, such as magnesium and aluminum, practically always exhibit burning mechanisms influenced by product condensation. In the presence of product condensation, there are a number of possible modes of quasisteady burning. Condensed products may accumulate on the surface of the particle, may accumulate in a shell at a reaction sheet located at some distance from the surface of the particle, may accumulate in a shell at a condensation sheet located outside a thin primary gas-phase reaction sheet, or may flow and diffuse to infinity in the form of fine particles. The last of these processes may be enhanced by thermophoretic motion (see Section E.2.5) of fine particles away from the hottest reaction zone under the influence of the temperature gradient [55]. Many theoretical analyses of the various types of combustion processes have been published [55]-[62]. Law s models [60], [61] of different... 

For the case where both reactants melt in the preheating zone and the liquid product forms in the reaction zone, a simple combustion model using the reaction cell geometry presented in Fig. 20d was developed by Okolovich et al. (1977). After both reactants melt, their interdiffusion and the formation of a liquid product occur simultaneously. Numerical and analytical solutions were obtained for both kinetic- and diffusion-controlled reactions. In the kinetic-limiting case, for a stoichiometric mixture of reactants (A and B), the propagation velocity does not depend on the initial reactant particle sizes. For dififiision-controlled reactions, the velocity can be written as... 

At this point, a distinction should be made between cellular and finite difference/element models. The latter are finite approximations of continuous equations [e.g., Eq. (11)], with the implicit assumption that the width of the reaction zone is larger than other pertinent length scales (diffusion, heterogeneity of the medium, etc.). However, no such assumptions need to be made for cellular... 

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