Diffusion and Mass-transfer Effects

Of more interest mechanistically though is the liquid nature of the molten phase present in the interstices of the inert porous support. It is now recognized that, deliberately or accidentally, a number of heterogeneously catalysed processes involve a supported liquid phase (SLP) rather than a solid catalyst. SLP catalysts are reviewed by Villadsen and Livbjerg. These include the vanadium-based sulphuric acid catalysts and, of comparable antiquity, the Deacon catalysts for oxidizing hydrogen chloride, where mixtures of copper and potassium chlorides can form melts in the catalyst support under reaction conditions. Thus in addition to diflfusional restrictions arising from pellet pore structure any 

Three broad regimes are thus possible (i) reaction takes place on the surface of the liquid, (ii) reaction takes place homogeneously in the bulk of the liquid catalyst without diffusional restrictions, and (iii) reaction takes place homogeneously in the liquid catalyst, but is limited to a layer near the liquid surface because of diffusional effects. 

Reaction between a gas and a liquid normally involves absorption and physical solution of the gas followed by homogeneous reaction between the dissolved species. The problem of gas absorption with chemical reaction has been extensively studied and in such systems the observed rate of gas absorption will be a function of the chemical reaction rate, the diffusion of the dissolved gas in the liquid, and, possibly, the fluid dynamics of the system (the rate of surface renewal) if surface tension driven or other circulation effects occur. There is no evidence of these so far in the thin films employed in practical catalysts. Danckwerts and Astarita give comprehensive treatments of the subject of gas absorption accompanied by reaction. 

In regimes (i) or (ii), if the concentration of catalyst is constant, the formal rate equations are the same as that used by Mars and Van Krevelen to model partial oxidation reactions. Here we have  

B and D are two forms of the catalyst which in a redox process would be different valency states of a transition metal ion, or alternatively they could represent unstable complexes of the dissolved catalyst with the reactants A and E. 

---

The ultimate width of a peak is determined by the total amount of diffusion occurring during movement of the solute through the system, and on the rate of mass transfer between the two phases. These effects are shown diagrammatically in Figure 4.14. Both diffusion and mass transfer effects are inter-dependent and complex, being made up of a number of contributions from different sources. Because they are kinetic effects, their influence on efficiency is determined by the rate at which the mobile phase... 

Diffusion and mass transfer effects cause the dimensions of the separated spots to increase in all directions as elution proceeds, in much the same way as concentration profiles become Gaussian in column separations (p. 86). Multiple path, molecular diffusion and mass transfer effects all contribute to spreading along the direction of flow but only the first two cause lateral spreading. Consequently, the initially circular spots become progressively elliptical in the direction of flow. Efficiency and resolution are thus impaired. Elution must be halted before the solvent front reaches the opposite edge of the plate as the distance it has moved must be measured in order to calculate the retardation factors (Rf values) of separated components (p. 86). 

Diffusion and mass transfer effects cause the dimensions of the separated spots to increase in all directions as elution proceeds, in much the same way as concentration profiles become Gaussian in column separations (p. 82). Multiple path, molecular,diffusion and mass transfer effects all contribute to spreading along the direction of flow but only the first two cause lateral... 

To determine the kinetic parameters, we use the differential reactor. In this continuous flow system, the variation in concentration between the inlet and outlet of the reactor should be small and finite. The conversions should be around 5-10%. Under these conditions, the diffusive and mass transfer effects are avoided, assuring a kinetic regime for the determination of the kinetic parameters. Unlike the case of the batch system, the spatial time and consequently, the inlet flow and the mass or volume of the reactor are varied. Therefore, the reaction rate is directly determined. 

The work is organized in two parts in the first part kinetics is presented focusing on the reaction rates, the influence of different variables and the determination of specific rate parameters for different reactions both homogeneous and heterogeneous. This section is complemented with the classical kinetic theory and in particular with many examples and exercises. The second part introduces students to the distinction between ideal and non-ideal reactors and presents the basic equations of batch and continuous ideal reactors, as well as specih c isothermal and non-isothermal systems. The main emphasis however is on both qualitative and quantitative interpretation by comparing and combining reactors with and without diffusion and mass transfer effects, complemented with several examples and exercises. Finally, non-ideal and multiphase systems are presented, as well as specific topics of biomass thermal processes and heterogeneous reactor analyses. The work closes with a unique section on the application of theory in laboratory practice with kinetic and reactor experiments. 

Examination of the parameters in the van Deemter expression (Equation 2.44), term by term, provides a basis for optimizing a packed column separation. The plate height, h, of a packed column may be represented as the sum of the eddy diffusion, molecular diffusion and mass transfer effects. Thus, to attain maximum column efficiency, each term in the plate height equation should be minimized ... 

This equilibrium based analytical theory of a conventional PSA process predicts that the mole fraction of species A in the high-pressure product gas, y, will continue to decrease with time (Shendalman and Mitchell, 1972 Chan et al., 1981). This is contrary to experimental observations. More exact theories based on axial diffusion and mass-transfer effects are needed to predict the observed behavior. 

Figure 1 illustrates the s)unmetrical nature of a chromatographic peak and S5urunetrical broadening. Figure 2 illustrates the mutiple-path, longitudinal diffusion and mass transfer effects. 

The total column dispersion is due to the combined effects of flow dispersion, longitudinal diffusion and mass transfer. 

This study was carried out to simulate the 3D temperature field in and around the large steam reforming catalyst particles at the wall of a reformer tube, under various conditions (Dixon et al., 2003). We wanted to use this study with spherical catalyst particles to find an approach to incorporate thermal effects into the pellets, within reasonable constraints of computational effort and realism. This was our first look at the problem of bringing together CFD and heterogeneously catalyzed reactions. To have included species transport in the particles would have required a 3D diffusion-reaction model for each particle to be included in the flow simulation. The computational burden of this approach would have been very large. For the purposes of this first study, therefore, species transport was not incorporated in the model, and diffusion and mass transfer limitations were not directly represented. 

Effects of diffusion and mass transfer on peak width, (a) Concentration profiles of a solute at the beginning of a separation, (b) Concentration profiles of a solute after passing some distance through the system. 

Surface effects and adsorption equilibria thus will significantly influence the course of photoelectrochemical transformations since they will effectively control the movement of reagents from the electrolyte to the photoactivated surface as well as the desorption of products (avoiding overreaction or complete mineralization). The stability and accessibility toward intermolecular reaction of photogenerated intermediates will also be controlled by the photocatalyst surface. Since diffusion and mass transfer to and from the photocatalyst surface will also depend on the solvent and catalyst pretreatment, detailed quantitative descriptions will be difficult to transfer from one experiment to another, although qualitative principles governing these events can be easily recognized. 

The log-mean-temperature-difference and effectiveness approaches are presented in heat-exchanger analysis since both are in wide use and each offers its own advantages to the designer. A brief introduction to diffusion and mass transfer is presented in order to acquaint the reader with these processes and to establish more firmly the important analogies between heat, mass, and momentum transfer. 

FIGURE 16 Schematic representation of the origins of zone-broadening behavior and mass transfer effects of a polypeptide or protein due to Brownian motion, eddy diffusion, mobile phase mass transfer, stagnant fluid mass transfer, and stationary-phase interaction transfer as the polypeptide or protein migrated through a column packed with porous particles of an interactive HPLC sorbent. 

We now discuss some of the main features of LLPTC models developed for reaction under neutral conditions. Evans and Palmer (1981) were among the first to consider the effect of diffusion and mass transfer inPTC. They considered PTC in liquid-liquid systems by considering two well-mixed bulk phases of uniform composition separated by a uniform stagnant mass-transfer layer at the interface, and set up equations for bulk phase species balance and mass conservation equations for simultaneous diffusion and reaction in the film. Dynamics of the interaction between reaction and diffusion were studied under these assumptions for two special cases (a) reaction which is pseudo-first-order in the quaternary ion-pair (b) mass-transfer controlled instantaneous reaction. 

Under these circumstances, the interparticle transport resistances can be neglected. What are left are the intraparticle resistances, i.e. the heat and mass transfer effects inside the catalyst particles. Since the current case reflects the situation that few reactant and product molecules exist in an environment of solvent molecules, the simplest Fick s law approach with effective diffusion coefficients can be considered as sufficient for the description of molecular diffusion. 

---