Uniform reaction model

In Figure 9.1(c), the opposite extreme case of a very porous solid B is shown. In this case, there is no internal diffusional resistance, all parts of the interior of B are equally accessible to A, and reaction occurs uniformly (but not instantaneously) throughout the particle. The concentration profiles are flat with respect to radial position, but cB decreases with respect to time, as indicated by the arrow. This model may be called a uniform-reaction model (URM). Its use is equivalent to that of a homogeneous model, in which the rate is a function of the intrinsic reactivity of B (Section 9.3), and we do not pursue it fiirther here. 

The biofilm thickness (Lf) and density (X = 50 g/L) were assumed uniform and the biofilm treated as a continuum. A substrate diffusion-reaction model assuming spherical particle was used. Diffusion coefficient of phenol and oxygen in the biofilm were assessed according to Fan et al. [64] ... 

The second class of solid reactions involves situations where the solid does not disappear or appear but rather transforms from one solid phase into another as the reaction proceeds, as shown in Figure 9-6. For transformations of solids there are several models that may be appropriate, depending on the microstmcture of the reacting solid. Limiting cases of concentration profiles within the solid are (1) uniform reaction and (2) film formation. Concentration profiles within the solid for these situations are shown in Figure 9-7. 

The solution found when the rate equations are pul equal to zero corresponds to equilibrium in the case of a uniform reaction environment, but also characterizes the steady state if it is assumed that the linear lattice separates two two-dimensional spaces such that on the one side the reaction is all 0 —> 1 according to ku k2, and k3 and on the other all 1 —> 0 according to k2 k2 and k3. As the k s can include functions of the environment within them such as the concentrations of a transported substance with which the lattice reacts, this model can be used to discuss transport through membranes with reactions governed by near neighbor effects. It will be clear that the reactivity of the linear lattice must be defined in an asymmetric fashion in order to obtain transport. 

Due to the pulsed action of balls on the reacting mixture, the process non-uniformity as in the time and in the space of a single mechanical action (the existence of gradient from the center of collision to its periphery), and changes of the conditions for chemical interaction in the course of activation, kinetic description of mechanochemical reactions is a complicated task. Therefore, one should not expect the creation of some universal reaction model only some particular models are possible. 

The uniform layer model of deposition has been implicit since Sato [3] and Newson [11]. This model leads to difficulties for a support surface of about 200 square meters per gram. A 20 weight % coke would occupy about 400 square meters, and 20 weight % vanadium pentasulfide would occupy 200 square meters. After a few months of operations, there would be 5 to 6 monolayer equivalent of deposits on the surface, so that the original cobalt-molybdenum surface would be completely covered. The remaining catalyst activity must be attributed to the activities of nickel and vanadium, which is perhaps ten times lower for the HDS reaction. 

B = 1. In this case, the reaction follows the uniform conversion model [31], which indicates that it proceeds under chemical reaction control, and the reactive agent first diffuses through the pore network and later reacts at any internal active site. The particle will retain its initial size during the conversion... 

A mechanism and models are proposed for the distribution and interaction of metals in EAH-USY zeolite. V was easily distributed in the zeolite and obeyed a uniform distribution model, while Ca possibly followed a non-uniform distribution model. The interaction between metals and the EAH-USY zeolite was caused by the chemical reaction of metals with A1 in the zeolite. Then, in various ways, depending on the type of metals, the metals destroy the zeolite structure. After reacting with Al, V tended to react with Si to affect the structure, while Ni continuously reacted with framework Al of zeolite. 

These data remain consistent with the gas diffusion reaction model. Thus, in the area of a deep and narrow trench, an adequate resupply of the silicon gas species to the deposition surfaces within the trench is not met as readily as on the top of the wafer s surface. Inevitably, the silicon gas specie will tend -because of its proximity - to deposit on the upper portion of the trench sidewalls. Hence, the growth rate near the top of the trench is faster than further down towards the bottom, a condition prone for eventual physical overgrowth (or closure) and void formation. A significant lowering of the vapor phase ( /Si ratio tends to cause general deposition problems in terms of growth rates (very low) and poor uniformities for typical batch sizes. Consequently, this approach is not fruitful. 

Reversible Reaction Model. In an ideal, contlnuous-stlrred extractor, the solute concentration of the bulk phase is uniform and equal to the outlet concentration, C b The solute concentration in the membrane portion of the globule, 0, is given as (39-41) ... 

Tuncel, A. 1999. A diffusion-reaction model for a-chymotrypsin carrying uniform thermosensitive gel beads. JAppl Polym Sci. lA 1025-1034. 

The use of LMPA intrusion along with visual analysis techniques, such as serial sectioning or microtomography, can provide a clear insight into the structural characteristics of porous media such as catalyst supports. Without the use of these visual analysis techniques, relying solely on mercury porosimetry and gas adsorption to derive a psd and subsequent diffiision-reaction models, a major error could be made, if the structure of the media is too spatially complex and non-uniformly variable. 

In an effort to find the optimum loading and performance of the monolith, the temperature and flow non-uniformities must be taken in to account. Therefore, a study was carried out firstly to harmonize the data reported in the literature for comparing the catalyst performances based on similar space times. Selected data from literature on the precious metals for CO oxidation reactions were harmonized and Arrhenius parameters were obtained (1-3,6-9). When the bed porosity data was not available an average estimate of the bed porosity was used to uniformly report the space times in terms of h units. The same analysis was applied to oxidation reactions of hydrocarbons present in the exhaust gas such as benzene, toluene, hexane and octane (3, 9-11). After the performance harmonization of the rate data on selected catalysts, the reaction model was used to estimate conversion profiles in a typical monolithic reactor. 

Kinetics models of gas-solid non-catalytic reaction include uniform conversion model (UCN), multiple fine particle model (GPM), crack core model (CCM), phase-change model (PCM), change void model (CVM), thermal decomposition model (TDM), shrinking core model with multi-step reactions, and multi-step reaction model of formation porous structure in reaction etc. Among these models, the shrinking core model (SCM) is the most important and most widely used. For conversion of solid it is also the most simple and practical model. Commonly it is suitable for experimental data. However, it can only be used in some reactions of many solid reactions. A more complex model must be used in other cases. 

Use of the information in these chapters will allow a researcher conducting experiments with catalysts in either an industrial or an academic laboratory to assess their results and determine the presence or absence of heat and mass transfer effects. Proper catalyst characterization provides the capability to report kinetic results properly in the form of specific or normalized activity, preferably in the form of a turnover frequency. The utilization and justification of reaction models based on uniform or ideal surfaces is discussed in detail, and numerous examples are provided. However, kinetic rate expressions based on the premise of nonuniform surfaces are also examined in depth to provide an alternate route to obtain a rate law, should the investigator wish to do so. In most studies of catalyzed reactions, the kinetics of these reactions lie at the heart of the investigation, not only because accurate comparisons of performance among different catalysts must be obtained, but also because accurate rate expressions can provide insight about the surface chemistry involved and they must be available for proper reactor design. 

Before the topic of nonuniform surfaces is concluded, it is interesting to compare the rate equation obtained by Temkin and Pyzhev for ammonia synthesis on iron (and discussed in Illustration 8.1) to one associated with a uniform surface using the same reaction model. Comparing only the forward rate in either sequence, one would have... 

---