Reaction diffusion prediction

Adsorption-Reaction and Reaction-Diffusion Predictions. Next, in order to investigate adsorption and reaction effects more fully, let us consider Z and Z p in normalized form. It is straightforward to show that... 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

Because of the strong dependence of composite properties on this final conversion, it is imperative that models of polymerizing systems be used to predict the dependence of the rate of polymerization and, hence, conversion on reaction conditions. The complexities of modeling such systems with autoacceleration, autodeceleration, and reaction diffusion all coupled with volume relaxation are enormous. However, several preliminary models for these systems have been developed [177,125,126,134-138]. These models are nearly all based on the coupled cycles illustrated in Fig. 5. 

In multifunctional monomer polymerizations, the mobility of radicals through segmental diffusion falls well before their mobility through reaction diffusion at very low functional group conversions (as compared to linear polymerizations). From this point in the reaction, the termination and propagation kinetic constants are found to be related, and the termination kinetic constant as a function of conversion may actually exhibit a plateau region. Figure 6 illustrates the typical behavior of kp and k, vs conversion as predicted by a kinetic based model. 

Relative viscosities are calculated from viscosities for the individual components at 0° (II7), weighting them on a mole fraction basis. The change in diffusivities and viscosities with temperature and pressure is assumed to be independent of gas mixture. If desired, more accurate calculations of diffusivities and viscosities of gas mixtures can be made using the approaches of Wilke (IIS) and Bromley and Wilke (II0), respectively. Table V presents relative values for Dfree, m, and p across the stagnant film for the gas-carbon reactions. Substituting these values in Equation (42), the relative reaction rates in Zone III for the gas-carbon reactions are calculated and also presented in Table V. Qualitatively, the rates of the carbon-oxygen and carbon-steam reactions are predicted to be about twice the rate... 

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

The catalyst intraparticle reaction-diffusion process of parallel, equilibrium-restrained reactions for the methanation system was studied. The non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components have been established, and solved using an orthogonal collocation method. The simulation values of the effectiveness factors for methanation reaction Ch4 and shift reaction Co2 are fairly in agreement with the experimental values. Ch4 is large, while Co2 is very small. The shift reaction takes place as direct and reverse reaction inside the catalyst pellet because of the interaction of methanation and shift reaction. For parallel, equilibrium-restrained reactions, effectiveness factors are not able to predict the catalyst internal-surface utilization accurately. Therefore, the intraparticle distributions of the temperature, the concentrations of species and so on should be taken into account. 

The experimentally-determined effectiveness factor is determined as the ratio of the experimental macro reaction rate to the intrinsic reaction rate under the same interface (bulk) composition and temperature. Based on the experimental conditions of the macrokinetics, the predicted effectiveness factors of the methanation reaction and the WGSR are obtained by solving the above non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components. Table 1 shows the calculated effectiveness factors and the experimental values. By... 

From a theoretical viewpoint, predicting the sequence of layer occurrence at the A-B interface would present no difficulties if the values of all the chemical constants entering a system of differential equations like (3.27) were known. For any multiphase binary system A-B, these values are determined by the physical-chemical properties of the elements A and B and their compounds. With their dependence on those properties established, the sequence of formation of compound layers would readily be predicted from the system of equations (3.27) or similar. Unfortunately, the theory of reaction diffusion has not yet reached this stage of its development. 

Nevertheless, in this book the number of the theoretically substantiated kinetic equations, for the experimentalist to use in practice, appears to exceed that resulting from purely diffusional considerations. Whether the experimentalist will be pleased with such an abundance of equations is a wholly different question. Still, for many researchers in the field it is so tempting to employ the only parabolic relation and then to discuss in detail the reasons for (unavoidable and predictable) deviations from its course. Note that unlike diffusional considerations where each interface is assumed to move according to the square root of the time, in the framework of the physicochemical approach the layer-growth kinetics are not predetermined by any additional assumptions, except basic ones, but immediately follow in a natural way from the proposed mechanism of the reaction-diffusion process. 

It appears relevant to note that many workers tend to overestimate the significance of thermodynamic predictions concerning the direction of the reaction-diffusion process. In fact, however, those only bear a likelihood character. Even if the free energy of formation of one compound from its constituents is -200 kJ mol-1, while that of the other is -20 kJ mol1, this does not necessarily mean, as often (tacitly or directly) assumed, that the former will occur first and the more so that its growth rate must be ten times greater than that of the latter. As exemplified with the growth rate of a compound layer in various diffusion couples of the same multiphase binary system, the opposite may well take place. 

In the case of thick samples (typically few mm thick), oxidation is restricted to superficial layers. As a result, O2 concentration in an elementary sublayer, located at a depth x beneath the sample surface, is all the more so small since this sublayer is deeper. The spatial distribution (in the sample thickness) of O2 concentration has been predicted from a balance equation expressing that [O2] variation in an elementary sublayer is equal to the O2 supply by diffusion (predicted by the classical Pick s second law) minus its consumption by the chemical reaction ... 

Effect of Temperature on Reaction Rate. The Klason lignin contents of red spruce residue extracted by methylamlne at four different temperatures along with the predicted curves from the reaction-diffusion model are shown In Figure 2. 

Figure 1. Comparison of the extent of delignificatlon predicted by the homogeneous (first-order, second-order and third-order) models and by the reaction-diffusion kinetic model. The open circles are the measured Klason lignin contents In the residues obtained from methylamine extraction of red spruce at 276 bar, 185 C and 1 g/mln solvent flow rate.

The bond-graph network of liquid membrane process can be successfully exploited for modeling the separation and transport ability of complex reaction-diffusion phenomena. However, such models involving appropriate mathematical formulations are especially useful in predicting the system s response to the changes in operating conditions and specific characteristics of the liquid membrane components. In general, such models are not... 

A disadvantage of ODE models is that they assume spatially homogeneous systems, an assumption that sometimes may lead to wrong predictions. Although in many cases spatial effects can be incorporated in the function I , there are situations where one may need to take into account diffusion and transport of proteins from one compartment to another. For the purpose, reaction-diffusion equations (RDEs) of the form... 

The coincidence of maxima in the methane oxidation rate and the sulfate reduction rate in Saanich Inlet strongly suggests that the methane oxidizing agent was sulfate, either via direct reaction, or coupled indirectly through reactions with other substrates (Devol, 1983). A methane-sulfate coupled reaction diffusion model was developed to describe the inverse relationship commonly observed between methane and sulfate concentrations in the pore waters of anoxic marine sediments. When fit to data from Saanich Inlet (B.C., Canada) and Skan Bay (Alaska), the model not only reproduces the observed methane and sulfate pore water concentration profiles but also accurately predicts the methane oxidation and sulfate reduction rates. In Saanich Inlet sediments, from 23 to 40% of the downward sulfate flux is consumed in methane oxidation while in Skan Bay this value is only about 12%. 

Up to now there is no general theory which allows to predict the parameters and the shape of a spiral wave in a two-dimensional excitable medium specified by a reaction-diffusion model like (9.1), though this problem has been the subject of numerous studies [14-16, 33]. 

The kinetic and solubility parameters obtained for the model were used together with the diffusional and textural parameters of the catalyst in a reaction-diffusion model in order to predict the performance of catalyst particles of industrial size. An agitated vessel operating in laboratory scale was considered, where the catalyst particles are placed in a basket and the gas phase is dispersed in the liquid phase, which is vigorously agitated. 

For the sake of comparison hydrogenation experiments with large cylindrical catalyst particles were carried out. The increase of the particle size diminished the velocity of catalytic hydrogenation. These experimental results provide a path for the process scale-up, i.e. a prediction of the hydrogenation rate on large catalyst particles starting from crushed particles. The values of the kinetic constants obtained for crushed particles were utilized and the ratio of porosity to tortuosity from the reaction-diffusion model was adjusted (0.167) to fit successfully the experimental data (Figure 10.40). 

During the oxidation of CO, CH4 and C3H8, the ignited state is characterized by a reaction front stabilized in a thin portion of the bed near the reactor inlet. This condition, corresponding to a diffusion-controlled reaction, is predicted by the known models of exothermic catalytic reactions [4], The chemical factors determining this dynamics are the heat of reaction and the activation energy. For all of the reactants considered in this study, a similar behaviour in the ignited state is observed. 

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