Diffusion reactant transport

The convolution treatment of the linear and semi-infinite diffusion reactant transport (Section 1.3.2) leads to the following relationship between the concentrations at the electrode surface and the current ... 

A mathematical model for DEFC was proposed by Pramanik and Basu describing different overpotentials [191]. The assumptions of their model are (i) the anode compartment considered as a well-mixed reactor, (ii) 1 bar pressure maintained both at the anode and cathode compartments, (iii) the transport processes are modelled in one dimension. The model accounts for Butler-Volmer-based descriptions of the ethanol electrooxidation mechanisms, diffusive reactants transport and ohmic losses at the electrode, current collector and electrode-current collector interfaces. The experiment data on current-voltage characteristics is predicted by the model with reasonable agreement and the influence of ethanol concentration and temperature on the performance of DEFC is studied by the authors (Fig. 8.19). 

In the film-penetration model (H19), it is assumed that the reactant A penetrates through the surface element by one-dimensional unsteady-state molecular diffusion. Convective transport is assumed to be insignificant. The diffusing stream of the reactant A is depleted along the path of diffusion by its reversible reaction with the reactant B, which is an existing component of the liquid surface element. If such a reaction can be represented as... 

If the reaction is diffusion controlled, i.e. the charge transfer is fast compared to reactant transport, the value of A i/2 is equal to k. ... 

A model of such structures has been proposed that captures transport phenomena of both substrates and redox cosubstrate species within a composite biocatalytic electrode.The model is based on macrohomo-geneous and thin-film theories for porous electrodes and accounts for Michaelis—Menton enzyme kinetics and one-dimensional diffusion of multiple species through a porous structure defined as a mesh of tubular fibers. In addition to the solid and aqueous phases, the model also allows for the presence of a gas phase (of uniformly contiguous morphology), as shown in Figure 11, allowing the treatment of high-rate gas-phase reactant transport into the electrode. 

Figure 13 shows the potential and concentration distributions for different values of dimensionless potential under conditions when internal pore diffusion (s = 0.1) and local mass transport (y = 10) are a factor. As expected the concentration and relative overpotential decrease further away from the free electrolyte (or membrane) due to the combined effect of diffusion mass transport and the poor penetration of current into the electrode due to ionic conductivity limitations. The major difference in the data is with respect to the variation in reactant concentrations. In the case when an internal mass transport resistance occurs (y = 10) the fall in concentration, at a fixed value of electrode overpotential, is not as great as the case when no internal mass transport resistance occurs. This is due to the resistance causing a reduction in the consumption of reactant locally, and thereby increasing available reactant concentration the effect of which is more significant at higher electrode overpotentials. 

Figure 6 shows the influence of the bulk diffusion coefficient, Dhi on the metal deposition profiles. Obviously, by decreasing the diffusivity the metal deposition process becomes more diffusion rate-determined. With decreasing diffusivity the transport of reactant and intermediates is decreased resulting in a less deep penetration into the catalyst pellet. Therefore, the metal deposition maximum is shifted further to the exterior of the catalyst pellet. 

Available reaction-transport models describe the second regime (reactant transport), which only requires material balances for CO and H2. Recently, we reported preliminary results on a transport-reaction model of hydrocarbon synthesis selectivity that describes intraparticle (diffusion) and interparticle (convection) transport processes (4, 5). The model clearly demonstrates how diffusive and convective restrictions dramatically affect the rate of primary and secondary reactions during Fischer-Tropsch synthesis. Here, we use an extended version of this model to illustrate its use in the design of catalyst pellets for the synthesis of various desired products and for the tailoring of product functionality and molecular weight distribution. 

Surprisingly, the synthesis rate is not influenced strongly by pellet diameter, in a size range where marked selectivity changes occur (Table V). This reflects reaction kinetics that are negative order in the diffusion-limited reactant (CO). These types of kinetics delay the inevitable decrease in catalyst productivity that ultimately accompanies reactant transport limitations and that occurs on Co catalysts only for larger and more active pellets. 

The probability of readsorption increases as the intrinsic readsorption reactivity of a-olefins (k,) increases and as their effective residence time within catalyst pores and bed interstices increases. The Thiele modulus [Eq. (15)] contains a parameter that contains only structural properties of the support material ( <>, pellet radius Fp, pore radius 4>, porosity) and the density of Ru or Co sites (0m) on the support surface. A similar dimensional analysis of Eqs. (l9)-(24), which describe reactant transport during FT synthesis, shows that a similar structural parameter governs intrapellet concentration gradients of CO and H2 [Eq. (25)]. In this case, the first term in the Thiele modulus (i/>co) reflects the reactive and diffusive properties of CO and H2 and the second term ( ) accounts for the effect of catalyst structure on reactant transport limitations. Not surprisingly, this second term is... 

The initial increase in C5+ selectivity as x increases arises from diffusion-enhanced readsorption of a-olefins. At higher values of CO transport restrictions lead to a decrease in C5+ selectivity. Because CO diffuses much faster than C3+ a-olefins through liquid hydrocarbons, the onset of reactant transport limitations occurs at larger and more reactive pellets (higher Ro, 0m) than for a-olefin readsorption reactions. CO transport limitations lead to low local CO concentrations and to high H2/CO ratios at catalytic sites. These conditions favor an increase in the chain termination probability (jSr, /Sh) and in the rate of secondary hydrogenation of a-olefins (j8s) and lead to lighter and more paraffinic products. 

The high asymptotic value of C5+ selectivity at large values of occurs on pellets that restrict the removal of reactive a-olefins and allow many readsorption events in the time required for intrapellet olefin removal by diffusion. Yet, transport restrictions within these pellets must not significantly hinder the rate of arrival of CO and H2 reactants to the active sites. Carbon number distributions also obey Flory kinetics for high values of because even the smaller olefins significantly react within a catalyst pellet. 

The route from reactant to product molecule in a monolith reactor comprises reactant transport from the bulk gas flow in a channel toward the channel wall, simultaneous diffusion and reaction inside the porous washcoat on the channel wall, and product transport from the wall back to the bulk flow of the gas phase. 

In this equation, represents the effective lateral diffusion/dispersion constant for laminar flow a value on the order of is suitable, and for turbulent flow the molecular diffusion coefficient in this expression should be replaced by the turbulent diffusion coefficient. Based on these simple relations it can be calculated that under typical conditions, lateral reactant transport takes place only over distances of a few subchannels. In other words, if the gas velocity through the wall channels differs much from the velocity through the central subchannels, the nonuniform flow profile can have a significant effect on the overall reactant conversion. In these situations the CBS model can be expected to give a better estimate of the reactor performance than the CB model. 

Fluid-fluid reactions are reactions that occur between two reactants where each of them is in a different phase. The two phases can be either gas and liquid or two immiscible liquids. In either case, one reactant is transferred to the interface between the phases and absorbed in the other phase, where the chemical reaction takes place. The reaction and the transport of the reactant are usually described by the two-film model, shown schematically in Figure 1.6. Consider reactant A is in phase I, reactant B is in phase II, and the reaction occurs in phase II. The overall rate of the reaction depends on the following factors (i) the rate at which reactant A is transferred to the interface, (ii) the solubihty of reactant A in phase II, (iii) the diffusion rate of the reactant A in phase II, (iv) the reaction rate, and (v) the diffusion rate of reactant B in phase II. Different situations may develop, depending on the relative magnitude of these factors, and on the form of the rate expression of the chemical reaction. To discern the effect of reactant transport and the reaction rate, a reaction modulus is usually used. Commonly, the transport flux of reactant A in phase II is described in two ways (i) by a diffusion equation (Pick s law) and/or (ii) a mass-transfer coefficient (transport through a film resistance) [7,9]. The dimensionless modulus is called the Hatta number (sometimes it is also referred to as the Damkohler number), and it is defined by... 

The solvent seems to favor diffusion and transport processes in the zeolite channels, the activation of the reactants is due to confinement catalysis. 

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