Mass transport factor

Supposing that heat and mass are transported by an identical convective mechanism, the catalyst temperature, given by eq 64, can also be expressed as a function of the Biot number for mass transport Bim. For this purpose, the equivalence of the heat and mass transport factors ( -factors) is utilized [19] ... 

A number of reviews have been published that provide quantitative discussions of the effect that these reaction parameters exert on the different steps of a three phase reaction.24-27 For simplification, however, a catalyzed reaction can be considered to be characterized by a kinetic rate factor, kr, and a mass transport factor, knv The relationship of these factors and the catalyst quantity, x, to the reaction rate is given by Eqn. 5.5 and its inverse, Eqn. 5.6.28... 

However, thermodynamic-based solubility data and its correlation with solvent properties (sub-H20) is only one facet in optimizing processes with sub-H20 - kinetic and mass transport factors must also be considered to receive maximum benefit in using this technique. Unfortunately, far less is known about the mass transport properties of solutes in sub-H20, but some useful correlations and experimental data do exist. These will be considered in the next section and how they can be utilized in conjunction with analyte solubility in optimizing and designing efficient experimental conditions. 

Apparent reaction rate constants indicating mass transport factors, and mass transport coefficients indicating diffusion processes from mineral surface to bulk solution, are introduced into an advection - diffusion - reaction model. 

Influence of the Kinetics of Electron Transfer on the Faradaic Current The rate of mass transport is one factor influencing the current in a voltammetric experiment. The ease with which electrons are transferred between the electrode and the reactants and products in solution also affects the current. When electron transfer kinetics are fast, the redox reaction is at equilibrium, and the concentrations of reactants and products at the electrode are those specified by the Nernst equation. Such systems are considered electrochemically reversible. In other systems, when electron transfer kinetics are sufficiently slow, the concentration of reactants and products at the electrode surface, and thus the current, differ from that predicted by the Nernst equation. In this case the system is electrochemically irreversible. 

The shape of a voltammogram is determined by several experimental factors, the most important of which are how the current is measured and whether convection is included as a means of mass transport. Despite an abundance of different voltam-metric techniques, several of which are discussed in this chapter, only three shapes are common for voltammograms (figure 11.33). 

The productivity of DR processes depeads oa chemical kinetics, as weU as mass and heat transport factors that combine to estabhsh the overall rate and extent of reduction of the charged ore. The rates of the reduction reactions are a function of the temperature and pressure ia the reductioa beds, the porosity and size distribution of the ore, the composition of the reduciag gases, and the effectiveness of gas—sohd contact ia the reductioa beds. The reductioa rate geaerahy iacreases with increasing temperature and pressure up to about 507 kPa (5 atm). 

The result is shown in Figure 10, which is a plot of the dimensionless effectiveness factor as a function of the dimensionless Thiele modulus ( ), which is R.(k/Dwhere R is the radius of the catalyst particle and k is the reaction rate constant. The effectiveness factor is defined as the ratio of the rate of the reaction divided by the rate that would be observed in the absence of a mass transport influence. The effectiveness factor would be unity if the catalyst were nonporous. Therefore, the reaction rate is... 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

The mass transport influence is easy to diagnose experimentally. One measures the rate at various values of the Thiele modulus the modulus is easily changed by variation of R, the particle size. Cmshing and sieving the particles provide catalyst samples for the experiments. If the rate is independent of the particle size, the effectiveness factor is unity for all of them. If the rate is inversely proportional to particle size, the effectiveness factor is less than unity and

experimental points allow triangulation on the curve of Figure 10 and estimation of Tj and ( ). It is also possible to estimate the effective diffusion coefficient and thereby to estimate Tj and ( ) from a single measurement of the rate (48). 

Intraparticle mass transport resistance can lead to disguises in selectivity. If a series reaction A — B — C takes place in a porous catalyst particle with a small effectiveness factor, the observed conversion to the intermediate B is less than what would be observed in the absence of a significant mass transport influence. This happens because as the resistance to transport of B in the pores increases, B is more likely to be converted to C rather than to be transported from the catalyst interior to the external surface. This result has important consequences in processes such as selective oxidations, in which the desired product is an intermediate and not the total oxidation product CO2. 

Rates and selectivities of soHd catalyzed reactions can also be influenced by mass transport resistance in the external fluid phase. Most reactions are not influenced by external-phase transport, but the rates of some very fast reactions, eg, ammonia oxidation, are deterrnined solely by the resistance to this transport. As the resistance to mass transport within the catalyst pores is larger than that in the external fluid phase, the effectiveness factor of a porous catalyst is expected to be less than unity whenever the external-phase mass transport resistance is significant, A practical catalyst that is used under such circumstances is the ammonia oxidation catalyst. It is a nonporous metal and consists of layers of wire woven into a mesh. 

The rate of mass transport is the product of these two factors, the density of atoms and the diffusion coefficient per atom, as shown in Fig. 6. Over a large temperature interval up to the mass transport coefficient is almost perfectly Arrhenius in nature. The enhanced adatom concentrations at high temperatures are offset by the lower mobility of the interacting atoms. Thus, surface roughening does not appear to cause anomalies in the... 

The ratio of the observed reaction rate to the rate in the absence of intraparticle mass and heat transfer resistance is defined as the elFectiveness factor. When the effectiveness factor is ignored, simulation results for catalytic reactors can be inaccurate. Since it is used extensively for simulation of large reaction systems, its fast computation is required to accelerate the simulation time and enhance the simulation accuracy. This problem is to solve the dimensionless equation describing the mass transport of the key component in a porous catalyst[l,2]... 

We shall now introduce an efficiency factor, which is again defined as the ratio of the conversion in the pore with and without mass transport limitation ... 

Inspection of Fig. 15.3 reveals that while for jo 0.1 nAcm , the effectiveness factor is expected to be close to 1, for a faster reaction with Jo 1 p,A cm , it will drop to about 0.2. This is the case of internal diffusion limitation, well known in heterogeneous catalysis, when the reagent concentration at the outer surface of the catalyst grains is equal to its volume concentration, but drops sharply inside the pores of the catalyst. In this context, it should be pointed out that when the pore size is decreased below about 50 nm, the predominant mechanism of mass transport is Knudsen diffusion [Malek and Coppens, 2003], with the diffusion coefficient being less than the Pick diffusion coefficient and dependent on the porosity and pore stmcture. Moreover, the discrete distribution of the catalytic particles in the CL may also affect the measured current owing to overlap of diffusion zones around closely positioned particles [Antoine et ah, 1998]. 

We begin by reviewing the principles of SECM methods, and present an overview of the instrumentation needed for experimental studies. A major factor in the success of SECM, in quantitative applications, has been the parallel development of theoretical models for mass transport. A detailed treatment of the theory for the most common SECM modes that have been used to study liquid interfaces is therefore given, along with key results from these models. A comprehensive assessment of the applications of SECM is provided and the prospects for the future developments of the methodology are highlighted. 

Much LC-MS work is carried out in a qualitative or semi-quantitative mode. Development of quantitative LC-MS procedures for polymer/additive analysis is gaining attention. When accurate quantitation is necessary, it is important to understand in depth the experimental factors which influence the quantitative response of the entire LC-MS system. These factors, which include solvent composition, solvent flow-rate, and the presence of co-eluting species, exert a major influence on analyte mass transport and ionisation efficiency. Analyte responses in MS procedures can be significantly affected by the nature of the organic modifier used in the RPLC... 

Rigorous calibration is a requirement for the use of the side-by-side membrane diffusion cell for its intended purpose. The diffusion layer thickness, h, is dependent on hydrodynamic conditions, the system geometry, the spatial configuration of the stirrer apparatus relative to the plane of diffusion, the viscosity of the medium, and temperature. Failure to understand the effects of these factors on the mass transport rate confounds the interpretation of the data resulting from the mass transport experiments. 

The intent of this chapter is to establish a comprehensive framework in which the physicochemical properties of permeant molecules, hydrodynamic factors, and mass transport barrier properties of the transcellular and paracellular routes comprising the cell monolayer and the microporous filter support are quantitatively and mechanistically interrelated. We specifically define and quantify the biophysical properties of the paracellular route with the aid of selective hydrophilic permeants that vary in molecular size and charge (neutral, cationic, anionic, and zwitterionic). Further, the quantitative interrelationships of pH, pKa, partition... 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

Thus, any factor that affects the solubility or apparent solubility of a drug entity has the potential of affecting the dissolution and diffusion mass transport of the molecule as shown in Eq. (6) [or Eq. (1)]. Polymorphism is one such important... 

Hydration can be an important factor in diffusion and mass transport phenomena in pharmaceutical systems. It may alter the apparent solubility or dissolution rate of the drug, the hydrodynamic radii of permeants, the physicochemical state of the polymeric membrane through which the permeant is moving, or the skin permeability characteristics in transdermal applications. 

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