Modeling diffusion with reaction

Modeling Diffusion with Chemical Reaction External Resistance to Mass Transfer 771... 

The Fickian diffusion models with constant effective diffusivities presented earlier and the rigorous dusty gas model presented in this section are not the only alternatives for modelling diffusion and reaction in porous catalyst pellets. Fickian models with effective diffusion coefficients which are varying with the change of concentration of the gas mixture can also be used. This is certainty more accurate compared with the Fickian model with constant diffusivities although of course less accurate than the dusty gas model. The main problem with these models is the development of relations for the change of diffusivities with the concentration of the gas mixture without solving the dusty gas model equations. Two such techniques are presented in this section and their results are compared with dusty gas model results. 

Y. Cheng, J. K. Suen, Radi-Z., S. D. Bond, M. ). Holst, and J. A. McCammon. Continuum simulations of acetylcholine diffusion with reaction-determined boundaries in neuromuscular junction models. Biophys. Chetn., 127[3) 129-139,2007. 

Combinations of weather conditions, wind speed and wind direction along witli boiling point, vapor density, diffusivity, and heat of vaporization of tlie chemical released vary the healtli impact of tlie released chemical on the nearby population. To model a runaway reaction, the release of 10,000 gallons was assumed to occur over a 15-minute period. Tlie concentration of the chemical released was estimated, using procedures described in Part III (Chapter 12) for each combination of weather condition, wind speed, and wind direction. The results, combined with population data for tlie area adjacent to tlie plant, led to probability estimates of the number of people affected. Table 21.5.3 sunimarizes tlie findings. 

Fig. 6. Film model for diffusion with simultaneous irreversible first-order chemical reaction [after Lightfoot (L5)].

Solutions for diffusion with and without chemical reaction in continuous systems have been reported elsewhere (G2, G6). In general, all the parameters in this model can be determined or estimated, and the theoretical expressions may assist in the interpretation of mass-transfer data and the prediction of equipment performance. 

A detailed description of AA, BB, CC step-growth copolymerization with phase separation is an involved task. Generally, the system we are attempting to model is a polymerization which proceeds homogeneously until some critical point when phase separation occurs into what we will call hard and soft domains. Each chemical species present is assumed to distribute itself between the two phases at the instant of phase separation as dictated by equilibrium thermodynamics. The polymerization proceeds now in the separate domains, perhaps at differen-rates. The monomers continue to distribute themselves between the phases, according to thermodynamic dictates, insofar as the time scales of diffusion and reaction will allow. Newly-formed polymer goes to one or the other phase, also dictated by the thermodynamic preference of its built-in chain micro — architecture. 

Models of chemical reactions of trace pollutants in groundwater must be based on experimental analysis of the kinetics of possible pollutant interactions with earth materials, much the same as smog chamber studies considered atmospheric photochemistry. Fundamental research could determine the surface chemistry of soil components and processes such as adsorption and desorption, pore diffusion, and biodegradation of contaminants. Hydrodynamic pollutant transport models should be upgraded to take into account chemical reactions at surfaces. 

Figure 5. Cartoon models of the reaction of methanol with oxygen on Cu(llO). 1 A methanol molecule arrives from the gas phase onto the surface with islands of p(2xl) CuO (the open circles represent oxygen, cross-hatched are Cu). 2,3 Methanol diffuses on the surface in a weakly bound molecular state and reacts with a terminal oxygen atom, which deprotonates the molecule in 4 to form a terminal hydroxy group and a methoxy group. Another molecule can react with this to produce water, which desorbs (5-7). Panel 8 shows decomposition of the methoxy to produce a hydrogen atom (small filled circle) and formaldehyde (large filled circle), which desorbs in panel 9. The active site lost in panel 6 is proposed to be regenerated by the diffusion of the terminal Cu atom away from the island in panel 7.

Detailed quantitative analyses of the data allowed the production of a mathematical model, which was able to reproduce all of the characteristics seen in the experiments carried out. Comparing model profiles with the data enabled the diffusion coefficients of the various components and reaction rates to be estimated. It was concluded that oxygen inhibition and latex turbidity present real obstacles to the formation of uniformly cross-linked waterborne coatings in this type of system. This study showed that GARField profiles are sufficiently quantitative to allow comparison with simple models of physical processes. This type of comparison between model and experiment occurs frequently in the analysis of GARField data. 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

Figure 9.7 shows concentration profiles schematically for A and B according to the two-film model. Initially, we ignore the presence of the gas film and consider material balances for A and B across a thin strip of width dx in the liquid film at a distance x from the gas-liquid interface. (Since the gas-film mass transfer is in series with combined diffusion and reaction in the liquid film, its effect can be added as a resistance in series.)... 

The half-order of the rate with respect to [02] and the two-term rate law were taken as evidence for a chain mechanism which involves one-electron transfer steps and proceeds via two different reaction paths. The formation of the dimer f(RS)2Cu(p-O2)Cu(RS)2] complex in the initiation phase is the core of the model, as asymmetric dissociation of this species produces two chain carriers. Earlier literature results were contested by rejecting the feasibility of a free-radical mechanism which would imply a redox shuttle between Cu(II) and Cu(I). It was assumed that the substrate remains bonded to the metal center throughout the whole process and the free thiyl radical, RS, does not form during the reaction. It was argued that if free RS radicals formed they would certainly be involved in an almost diffusion-controlled reaction with dioxygen, and the intermediate peroxo species would open alternative reaction paths to generate products other than cystine. This would clearly contradict the noted high selectivity of the autoxidation reaction. 

The fact that diffusion models describe a number of chemical processes in solid particles is not surprising since in most cases, mass transfer and chemical kinetics phenomena occur simultaneously and it is difficult to separate them [133-135]. Therefore, the overall kinetics of many chemical reactions in soils may often be better described by mass transfer and diffusion-based models than with simple models such as first-order kinetics. This is particularly true for slower chemical reactions in soils where a fast reaction is followed by a much slower reaction (biphasic kinetics), and is often observed in soils for many reactions involving organic and inorganic compounds. 

Gifford and Hanna tested their simple box model for particulate matter and sulfur dioxide predictions for annual or seasonal averages against diffusion-model predictions. Their conclusions are summarized in Table 5-3. The correlation coefficient of observed concentrations versus calculated concentrations is generally higher for the simple model than for the detailed model. Hanna calculated reactions over a 6-h period on September 30, 1%9, with his chemically reactive adaptation of the simple dispersion model. He obtained correlation coefficients of observed and calculated concentrations as follows nitric oxide, 0.97 nitrogen dioxide, 0.05 and rhc, 0.55. He found a correlation coefficient of 0.48 of observed ozone concentration with an ozone predictor derived from a simple model, but he pointed out that the local inverse wind speed had a correlation of 0.66 with ozone concentration. He derived a critical wind speed formula to define a speed below which ozone prediction will be a problem with the simple model. Further performance of the simple box model compared with more detailed models is discussed later. 

The beginning of modeling of polymer-electrolyte fuel cells can actually be traced back to phosphoric-acid fuel cells. These systems are very similar in terms of their porous-electrode nature, with only the electrolyte being different, namely, a liquid. Giner and Hunter and Cutlip and co-workers proposed the first such models. These models account for diffusion and reaction in the gas-diffusion electrodes. These processes were also examined later with porous-electrode theory. While the phosphoric-acid fuel-cell models became more refined, polymer-electrolyte-membrane fuel cells began getting much more attention, especially experimentally. 

The equations used in these models are primarily those described above. Mainly, the diffusion equation with reaction is used (e.g., eq 56). For the flooded-agglomerate models, diffusion across the electrolyte film is included, along with the use of equilibrium for the dissolved gas concentration in the electrolyte. These models were able to match the experimental findings such as the doubling of the Tafel slope due to mass-transport limitations. The equations are amenable to analytic solution mainly because of the assumption of first-order reaction with Tafel kinetics, which means that eq 13 and not eq 15 must be used for the kinetic expression. The different equations and limiting cases are described in the literature models as well as elsewhere. 

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