Reaction-diffusion modeling

Chapter 8 describes a number of generalized CA models, including reversible CA, coupled-map lattices, quantum CA, reaction-diffusion models, immunologically motivated CA models, random Boolean networks, sandpile models (in the context of self-organized criticality), structurally dynamic CA (in which the temporal evolution of the value of individual sites of a lattice are dynamically linked to an evolving lattice structure), and simple CA models of combat. 

Huang, D.S., Huang, K.L., Chen, S.P. et al. (2008) Rapid reaction-diffusion model for the enantioseparation of phenylalanine across hollow fiber supported liquid membrane. Separation Science and Technology, 43 (2), 259-272. 

Schwann cells incorporated in a collagen matrix and injected into PLLA conduits were found to demonstrate comparable SFI values compared to isograft controls, but showed a statistically lower number of axons for both the high and low density Schwann cells groups and the collagen samples compared to the isograft controls (Evans et al., 2002). These results can be explained by a simple reaction diffusion model (Rutkowski and Heath, 2002b) described earlier. 

Figure 7. Mo concentotions (o) and isotopic compositions ( ) from reducing pore fluids in Santa Monica Basin (McManus et al. 2002). Dotted line indicates seawater values for both variables. The data can be fit by a 1-D reaction-diffusion model wifli a fractionation factor of —1.005. The effective fractionation factor for Mo removal across tiie sediment-water interface is smaller, <1.0025 (see text).

Experimental data for the interligand electron transfer kinetics following photoexcitation of [Os(bpy)3] " " are in agreement with a reaction/diffusion model measurements were made in a range of solvents. The variable parameters in the model are interligand electronic coupling and solvent polarization barrier height. 

We have presented a general reaction-diffusion model for porous catalyst particles in stirred semibatch reactors applied to three-phase processes. The model was solved numerically for small and large catalyst particles to elucidate the role of internal and external mass transfer limitations. The case studies (citral and sugar hydrogenation) revealed that both internal and external resistances can considerably affect the rate and selectivity of the process. In order to obtain the best possible performance of industrial reactors, it is necessary to use this kind of simulation approach, which helps to optimize the process parameters, such as temperature, hydrogen pressure, catalyst particle size and the stirring conditions. 

This alternation between stable and unstable states follows that found for the CSTR. Other responses such as slowing down described in chapter 8 also occur for the reaction-diffusion model. 

Now let us return to the reaction-diffusion model and imagine that the stirring is somehow stopped. The question we address is whether the region of instability can be extended beyond the closed region of Fig. 10.3, particularly when the participants diffuse at different rates so ft = 1. 

Arcuri, P. and Murray, J. D. (1986). Pattern sensitivity to boundary and initial conditions in reaction-diffusion models. J. Math. Biol., 24, 141-65. 

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. 

Bessler W., 2005. New computational approach for SOFC impedance from detailed electrochemical reaction-diffusion models. Solid State Ionics 176, 997-1011. 

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. 

Keywords Catalyst Parallel reactions Equilibrium-restrained Reaction-diffusion model... 

Commonly [17], when the length-to-diameter ratio of a cylindrical catalyst is close to 1, the cylindrical catalyst can be simplified as a sphere, the radius of which, Rp, is calculated by 3 Kp/.S p. The one-dimensional, key-component based reaction-diffusion models of methanation system are as follows ... 

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... 

Figure 12. Equivalence between the reaction/diffusion model and the IEM model for second-order consecutive competing reactions...

Spatially distributed systems and reaction-diffusion modeling... 

Many biochemical signaling processes involve the coupled reaction diffusion of two or more substrates. Metabolic biochemical pathways are mainly multicomponent reaction cycles leading to binding and/or signaling and are coupled to the transport of substrates. A reaction-diffusion model can also describe the diffusion of certain proteins along the bacterium and their transfer between the cytoplasmic membrane and cytoplasm, and the generation of protein oscillation along the bacterium (Wood and Whitaker, 2000). 

Example 12.11 Reaction-diffusion model The linear stability analysis (Zhu and Li, 2002) may be used to investigate the evolution of a reaction-diffusion model of solid-phase combustion (Feng et al., 1996). The diffusion coefficients of the oxygen and magnesium (g) are the two controlling parameters besides kinetics... 

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