Limiting diffusion resistance

Chen et al. [54] have reported a model for the assessment of the combined effects of the intrinsic reaction kinetics and dye diffusion into phosphorylated polyvinyl alcohol (PVA) gel beads. The analysis of the experimental data in terms of biofilm effectiveness factor highlighted the relevance of intraparticle diffusion to the effective azo-dye conversion rate. On the basis of these results, they have identified the optimal conditions for the gel bead diameter and PVA composition to limit diffusion resistance. 

Now let us consider the exterior problem about mass exchange between a spherical drop (bubble) of radius a and a translational Stokes flow with limiting diffusion resistance of the continuous phase. 

The asymptotic solution ( - large) for tj is [2/(n + l)]1/2/, of which the result given by 8.5-14c is a special case for a first-order reaction. The general result can thus be used to normalize the Thiele modulus for order so that the results for strong pore-diffusion resistance all fall on the same limiting straight line of slope - 1 in Figure 8.11. The normalized Thiele modulus for this purpose is... 

The last part of the polarization curve is dominated by mass-transfer limitations (i.e., concentration overpotential). These limitations arise from conditions wherein the necessary reactants (products) cannot reach (leave) the electrocatalytic site. Thus, for fuel cells, these limitations arise either from diffusive resistances that do not allow hydrogen and oxygen to reach the sites or from conductive resistances that do not allow protons or electrons to reach or leave the sites. For general models, a limiting current density can be used to describe the mass-transport limitations. For this review, the limiting current density is defined as the current density at which a reactant concentration becomes zero at the diffusion medium/catalyst layer interface. 

Figure 18.7 Shows the limits for negligible and for strong pore diffusion resistance.

The fluid resistance experienced by a macromolecular solute moving in dilute solution depends on the shape and size of the molecule. A number of physical quantities have been introduced to express this. Typical ones are intrinsic viscosity [ry], limiting sedimentation coefficient s0, and limiting diffusion coefficient D0. The first is related to the rotation of the solute, while the last two are concerned with the translational motion of the solute. A wealth of theoretical and experimental information about these hydrodynamic quantities is already available for randomly coiled chains (40, 60). However, the corresponding information on non-randomly coiled polymers is as yet rather limited in number and in variety. 

When immobilizing biocatalysts within polymer gels using physical entrapment methods, we may take advantage of the great resistance to the diffusion of macromolecular substances due to the gel porosity. However, this limited diffusion within the gel phase also causes a reduced mass transfer rate for low... 

The polarization curve is obtained step by step, at every potential until obtention of a steady-state value. The polarization curve must be identical during forward or backward potential scan. If not, either the steady state has not been obtained, or, more frequently, the surface of the electrode has been modified by the electrochemical reaction. Covering the platinum electrode by a Nafion film reduces the limiting current 71( by the addition of a supplementary diffusion resistance, depending on the thickness of the Nafion film (Figures 1.13 and 1.14). 

Von Getler et al. [23] were the first to consider these processes during spray absorption drying when they examined the material system of SO2 and Ca(OH)2. In the opinion of these authors, the absorption of sulfur dioxide is limited either by the dissolution of solid or by the gas-phase mass transfer in the first drying period of the drop. The product of this reaction causes substantial diffusion resistance for the absorbed sulfur dioxide and thus obstructs further reactions as the calcium hydroxide remains in the core. 

The support has an internal pore structure (i.e., pore volume and pore size distribution) that facilitates transport of reactants (products) into (out of) the particle. Low pore volume and small pores limit the accessibility of the internal surface because of increased diffusion resistance. Diffusion of products outward also is decreased, and this may cause product degradation or catalyst fouling within the catalyst particle. As discussed in Sec. 7, the effectiveness factor Tj is the ratio of the actual reaction rate to the rate in the absence of any diffusion limitations. When the rate of reaction greatly exceeds the rate of diffusion, the effectiveness factor is low and the internal volume of the catalyst pellet is not utilized for catalysis. In such cases, expensive catalytic metals are best placed as a shell around the pellet. The rate of diffusion may be increased by optimizing the pore structure to provide larger pores (or macropores) that transport the reactants (products) into (out of) the pellet and smaller pores (micropores) that provide the internal surface area needed for effective catalyst dispersion. Micropores typically have volume-averaged diameters of 50 to... 

In Figure 4.31, R1 is the charge-transfer resistance, CPE1 represents the double-layer capacitance in the form of a constant phase element, R2 is the adsorption resistance or the diffusion resistance due to slow gas diffusion through the limited porous structure, and CPE2 is the adsorption-related constant phase element or capacitance. In a tme case, neither the double-layer nor the adsorption-related capacitance is pure capacitance, and therefore it is more realistic to use CPE instead of capacitance. 

A few reactor models have recently been proposed (30-31) for prediction of integral trickle-bed reactor performance when the gaseous reactant is limiting. Common features or assumptions include i) gas-to-liquid and liquid-to-solid external mass transfer resistances are present, ii) internal particle diffusion resistance is present, iii) catalyst particles are completely externally and internally wetted, iv) gas solubility can be described by Henry s law, v) isothermal operation, vi) the axial-dispersion model can be used to describe deviations from plug-flow, and vii) the intrinsic reaction kinetics exhibit first-order behavior. A few others have used similar assumptions except were developed for nonlinear kinetics (27—28). Only in a couple of instances (7,13, 29) was incomplete external catalyst wetting accounted for. 

It plays the same role as the effectiveness factor in heterogeneous catalysis and is a measure of the film thickness uniformity. It represents the ratio of the total reaction rate on each pair of wafers to that we would obtain if the concentration in the cell formed by the two wafers were equal to the bulk concentration everywhere. Thus, if the surface reaction is the rate controlling step, n = 1, whereas if the diffusion between the wafers controls, n < 1. In the limit of strong diffusion resistance the deposition is confined to a narrow outer band of the wafers and a strongly nonuniform film results. 

A simplification is often employed for effectiveness factor calculations in the asymptotic limit of strong intraparticle diffusion resistance (12,13). In this situation, an alternative form of the key component mass balance can be written as follows ... 

In the other limiting case when h 2D/kML, the permeability is determined by the diffusion resistance of the monolayer... 

One of the substrates that deserves special attention in our discussion is silicon (Si), and in particular Si wafers for producing solar cells. A variety of sirrface treatments are applied to Si in the solar cell industry Native Si vs. chemically protected (SiN, for example), pohshed versus unpohshed versus patterned to give surface texture, etc. In order to ensirre proper Si-metal contacts, careful control of the conductive ink composition is required. In addition, very high curing temperatures are usually employed in the manufacture of solar cells, thus there is a need for a suitable binding component. Special care is taken to avoid excessive diffusion of metal into the emitter layer, nevertheless, somewhat limited diffusion is desirable so that contact resistance is adequate. ... 

For pore diffusion resistances in reactions having moderate heat evolution, the following phenomena characteristically hold true in industrial ammonia synthesis [212] in the temperature range in which transport limitation is operative, the apparent energy of activation falls to about half its value at low temperatures the apparent activation energy and reaction order, as well as the ammonia production per unit volume of catalyst, decrease with increasing catalyst particle size [211], [213]-[215]. For example at the gas inlet to a TVA converter, the effective rate of formation of ammonia on 5.7-mm particles is only about a quarter of the rate measured on very much smaller grains (Fig. 13) [157]. 

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