Diffusion mass transfer-limited reactions

The diffusion paths for heat and mass transfer are very small, making micro reactors ideal candidates for heat or mass transfer-limited reactions. 

The performance of adsorption processes results in general from the combined effects of thermodynamic and rate factors. It is convenient to consider first thermodynamic factors. These determine the process performance in a limit where the system behaves ideally i.e. without mass transfer and kinetic limitations and with the fluid phase in perfect piston flow. Rate factors determine the efficiency of the real process in relation to the ideal process performance. Rate factors include heat-and mass-transfer limitations, reaction kinetic limitations, and hydro-dynamic dispersion resulting from the velocity distribution across the bed and from mixing and diffusion in the interparticle void space. 

A measure of the absence of internal (pore diffusion) mass transfer limitations is provided by the internal effectiveness factor, t, which is defined as the ratio of the actual overall rate of reaction to the rate that would be observed if the entire interior surface were exposed to the reactant concentration and temperature existing at the exterior of the catalyst pellet. A value of 1 for rj implies that all of the sites are being utilized to their potential, while a value below, say, 0.5, signals that mass transfer is limiting performance. The value of rj can be related to that of the Thiele modulus, 4>, which is an important dimensionless parameter that roughly expresses a ratio of surface reaction rate to diffusion rate. For the specific case of an nth order irreversible reaction occurring in a porous sphere,... 

One of the most difficult decisions that a textbook writer has to make is to select what material to cover and what topics to leave out. This is especially difficult in chemical reaction engineering because of the wide scope of the field and the diversity of topics that it covers. As the title indicates, this book focuses on the analysis and design of chemical reactors. The objective of the book is to present a comprehensive, unified methodology to analyze and design chemical reactors that overcomes the deficiencies of the current pedagogy. To concentrate on this objective, some topics that are commonly covered in chemical reaction engineering textbooks (chemical kinetics, catalysis, effect of diffusion, mass-transfer limitation, etc.) are... 

Immobilization, dehned as the physical confinement or localization of an enzyme into a specihc micro-environment, has been a very common approach to prepare enzymes for aqueous as well as nonaqueous applications. For nonaqueous enzymol-ogy, immobilization improves storage and thermal stability, facilitates enzyme recovery, and enhances enzyme dispersion. In addition, immobilized enzymes are readily incorporated in packed bed bioreactors, allowing for continuous operation of reactions. Moreover, lyophilized enzyme powders often aggregate and attach to reactor walls, particularly when the water activity is moderately high. The major disadvantage of immobilization is low activity, induced by pore diffusion mass transfer limitations and by alteration of protein stmcture. For enzymes in nonaqueous media, the following broad categories of immobilization exist ... 

Closure. After completing this chapter, the reader should be able to define and describe molecular diffusion and how it varies with temperature and pressure, the molar flux, bulk flow, the mass transfer coefficient, the Sherwood and Schmidt numbers, and the correlations for the mass transfer coefficient. The reader should be able to choose the appropriate correlation and calculate the mass transfer coefficient, the molar flux, and the rate of reaction. The reader should be able to describe the regimes and conditions under which mass transfer-limited reactions occur and when reaction rate limited reactions occur and to make calculations of the rates of reaction and mass transfer for each case. One of die most imponant areas for the reader apply the knowledge of this (and other chapters) is in their ability to ask and answer "What if. , questions. Finally, the reader should be able to describe the shrinking core model and apply it to catalyst regeneration and pharmacokinetics. 

As in any solid-liquid reaction, when the solid is sparingly soluble, reaction occurs within the solid by diffusion of the liquid-phase reactant into it across the liquid film surrounding the solid. Thus two diffusion parameters are operative, the solid-liquid mass transfer coefficient sl and the effective diffusivity D. of the reactant in the solid. A reaction in the solid can occur by any of several mechanisms. The simpler and more common of these were briefly explained in Chapter 15. For reactions following the sharp interface model, ultrasound can enhance either or both these constants. Indeed, in a typical solid-liquid reaction such as the synthesis of dibenzyl sulfide from benzyl chloride and sodium sulfide ultrasound enhances SL by a factor of 2 and by a factor of 3.3 (Hagenson and Doraiswamy, 1998). Similar enhancement in was found for a Michael addition reaction (Ratoarinoro et al., 1995) and for another mass transfer-limited reaction (Worsley and Mills, 1996). 

Greater control over reactions. The diffusion paths for heat and mass transfer in microfluidic systems are very short, making such systems ideal candidates for heat- or mass-transfer-limited reactions. The surface-to-volume ratio of microscopic structures is very high. Thus, surface effects are likely to dominate over volumetric effects, increasing selectivity and yield. 

Prior to conducting the DOE (design of experiments) described in Table 3, it was established that no reaction took place in the absence of a catalyst and that the reactions were conducted in the region where chemical kinetics controlled the reaction rate. The results indicated that operating the reactor at 1000 rpm was sufficient to minimize the external mass-transfer limitations. Pore diffusion limitations were expected to be minimal as the median catalyst particle size is <25 pm. Further, experiments conducted under identical conditions to ensure repeatability and reproducibility in the two reactors yielded results that were within 5%. 

Many challenges remain to be addressed in this field. The use of immobilized catalysts can often reduce the activity of a catalyst Reduced reaction rates due to diffusion limitations through a permeable membrane capsule and the ease or practicality of the synthesis of these catalyst scaffolds are issues that may pose problems. In some cases, these issues have been resolved, but this is often at the expense of other properties of the capsule. For example, the use of thin walls to reduce mass transfer limitations can be at the expense of nanocapsule strength and stability. 

The flow terms represent the convective and diffusive transport of reactant into and out of the volume element. The third term is the product of the size of the volume element and the reaction rate per unit volume evaluated using the properties appropriate for this element. Note that the reaction rate per unit volume is equal to the intrinsic rate of the chemical reaction only if the volume element is uniform in temperature and concentration (i.e., there are no heat or mass transfer limitations on the rate of conversion of reactants to products). The final term represents the rate of change in inventory resulting from the effects of the other three terms. 

Slurry Reactors. Slurry reactors are commonly used in situations where it is necessary to contact a liquid reactant or a solution containing the reactant with a solid catalyst. To facilitate mass transfer and effective catalyst utilization, the catalyst is usually suspended in powdered or in granular form. This type of reactor has been used where one of the reactants is normally a gas at the reaction conditions and the second reactant is a liquid, e.g., in the hydrogenation of various oils. The reactant gas is bubbled through the liquid, dissolves, and then diffuses to the catalyst surface. Obviously mass transfer limitations can be quite significant in those instances where three phases (the solid catalyst, and the liquid and gaseous reactants) are present and necessary to proceed rapidly from reactants to products. 

The numerator of the right side of this equation is equal to the chemical reaction rate that would prevail if there were no diffusional limitations on the reaction rate. In this situation, the reactant concentration is uniform throughout the pore and equal to its value at the pore mouth. The denominator may be regarded as the product of a hypothetical diffusive flux and a cross-sectional area for flow. The hypothetical flux corresponds to the case where there is a linear concentration gradient over the pore length equal to C0/L. The Thiele modulus is thus characteristic of the ratio of an intrinsic reaction rate in the absence of mass transfer limitations to the rate of diffusion into the pore under specified conditions. 

The reaction will then appear to follow first-order kinetics, regardless of the functional form of the intrinsic rate expression and of the effectiveness factor. This first-order dependence is characteristic of reactions that are mass transfer limited. The term diffusion controlled is often applied to reactions that occur under these conditions. 

It can be observed that the initial rate of polymerization decreases and the autoacceleration peak is suppressed as the TED concentration is increased. The TED molecules generate dithiocarbamyl (DTC) radicals upon initiation. As a result, termination may occur by carbon-carbon combination which leads to a dead polymer and by carbon-DTC radical reaction which produces a reinitiatable ( living ) polymer. The cross-termination of carbon-DTC radicals occurs early in the reaction (with the carbon-carbon radical termination), and this feature is observed by the suppression of the initial rate of polymerization. As the conversion increases, the viscosity of the system poses mass transfer limitations to the bimolecular termination of carbon radicals. As has been observed in Figure 3, this effect results in a decrease in the ktCC. However, as the DTC radicals are small and mobile, the crosstermination does not become diffusion limited, i.e., the kinetic constant for termination of carbon-DTC radicals, ktCS, does not decrease. Therefore, the crosstermination becomes the dominant reaction pathway. This leads to a suppression of the autoacceleration peak as the carbon-DTC radical termination limits the carbon radical concentration to a low value, thus limiting the rate of polymerization. This observation is in accordance with results of previous studies (10) with XDT and TED, where it was found that when there was an excess of DTC radicals, the carbon radical concentration was lower and the cross-termination reaction was the dominant termination pathway. 

The chemical species were treated as passive scalar tracers in the unsteady LBM equations. The reaction was simulated as being mass-transfer limited at low Re — 166, with diffusivities in the ratios DA DB Dc— 1 3 2. The concentration fields shown in Fig. 16 are different for each species due to the different diffusivities. The slow-diffusing species A is transported mainly by convection and regions of high or low concentration correspond to features of the flow field. A more uniform field is seen for the concentration of faster... 

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. 

Figure 7-15 Plots of r versus T and log i versus 1/r. We expect the rate to exhibit breaks on the 1/r plot as the reaction process goes from reaction limited at low temperature, pore diffusion limited at intermediate temperature, and external mass transfer limited at high temperature.

There are a number of examples of tube waU reactors, the most important being the automotive catalytic converter (ACC), which was described in the previous section. These reactors are made by coating an extruded ceramic monolith with noble metals supported on a thin wash coat of y-alumina. This reactor is used to oxidize hydrocarbons and CO to CO2 and H2O and also reduce NO to N2. The rates of these reactions are very fast after warmup, and the effectiveness factor within the porous wash coat is therefore very smaU. The reactions are also eternal mass transfer limited within the monohth after warmup. We wUl consider three limiting cases of this reactor, surface reaction limiting, external mass transfer limiting, and wash coat diffusion limiting. In each case we wiU assume a first-order irreversible reaction. 

Figure 7-20 Sketch of tube wall reactor with a porous catalyst film of thickness t on walls. Expected reactant concentration profiles with reaction-limited, mass-transfer-limited, and pore-diffusion-limited reaction.

We have presented a general reaction-diffusion model for porous catalyst particles in stirred semibatch reactors applied to three-phase processes. The model was solved numerically for small and large catalyst particles to elucidate the role of internal and external mass transfer limitations. The case studies (citral and sugar hydrogenation) revealed that both internal and external resistances can considerably affect the rate and selectivity of the process. In order to obtain the best possible performance of industrial reactors, it is necessary to use this kind of simulation approach, which helps to optimize the process parameters, such as temperature, hydrogen pressure, catalyst particle size and the stirring conditions. 

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