Steps in series

The appearance of this heterogeneous form for the rate expression reflects the presence of a mass transfer step in series with the reaction step. If the parameter values are known, this ODE for bi i) can be integrated subject to the initial condition that bi=(bi)o at t = 0. The result can then be used to find a (f). 

A reaction network could then consist of two steps in series in which CH4 forms HCHO, which subsequently oxidizes to CO, together with a third step in parallel in which CH4... 

We can develop a simple shrinking-particle kinetics model by taking the two rate-processes involved as steps in series, in a treatment that is simpler than that used for the SCM, although some of the assumptions are the same ... 

The rate of a reaction, A = Products, is controlled by three steps in series,... 

We consider a simple reaction composed of two elementaiy steps in series with the affinity distribution ratio m - AgilAg and the stoichiometric niunbers vj and vj. The mean stoichiometric number v is then given by Eqn. 7-52 ... 

For multiple reactions a change in the observed activation energy with temperature indicates a shift in the controlling mechanism of reaction. Thus, for an increase in temperature Eq s rises for reactions or steps in parallel, Eobs falls for reactions or steps in series. Conversely, for a decrease in temperature E s falls for reactions in parallel, E s rises for reactions in series. These findings are illustrated in Fig. 2.3. 

Second, for steps in series if an early step needs a high temperature and a later step needs a low temperature, then a falling progression of temperatures should be used. Analogous arguments hold for other progressions. 

From Fig. E17.1 we see that two steps in series are involved—mass transfer of oxygen to the surface followed by reaction at the surface of the particle. 

Consider steps in series. In general, if all the steps are linear in concentration, then it is easy to combine them. However, if any of the steps are nonlinear, then you will get a messy overall expression. Therefore, you may try to bypass this nonlinear step in one of various ways. Approximating the versus curve by a first-order expression is probably the most useful procedure. 

At steady state the flow rate to the surface is equal to the reaction rate at the surface (steps in series). So... 

The ideas of reaction steps in series and rate-limiting steps can be best understood by the idea of resistors in series. If a voltage V is applied across a resistor with resistance R, the current flow 1 is given by Ohm s Law,... 

Figure 9-16 Sketch of concentration profiles and electrical voltage analog for reaction and diffusion steps in series. Also shown is the situation for parallel channels for a particular step that may occur through two alternate routes.

A fast step in series with a slow step... 

When a process consists of several individual steps in series, the concept of the ratecontrolling step simply states that one of the several steps involved provides the major resistance to the overall process. In such a case, this slow step is termed the ratecontrolling step and can be considered as acting alone (Levenspiel, 1972). 

In Equation 3.1, the suffix i usually designates a reaction product. Ihe rate r,-is negative, in case i is a reactant. Several factors, such as temperature, pressure, the concentrations of the reactants, and also the existence of a catalyst affect the rate of a chemical reaction. In some cases, what appears to be one reaction may in fact involve several reaction steps in series or in parallel, one of which may be rate limiting. 

Figu re 4.9 Thermogram obtained in a reaction calorimeter (Mettler-Toledo RC1) during the study of a catalytic hydrogenation. The heat release rate suddenly increases after 3 hours, which indicates a complex reaction scheme involving several steps in series. The second sharp peak after 5.8 hours is due to... 

The need for novel catalytic processes is clear and, as discussed in Chapter 9, combining catalytic steps into cascade processes, thus obviating the need for isolation of intermediate products, results in a further optimization of both the economics and the environmental footprint of the process. In vivo this amounts to metabolic pathway engineering [20] of the host microorganism (see Chapter 8) and in vitro it constitutes a combination of chemo- and/or biocatalytic steps in series and is referred to as cascade catalysis (see Chapter 9). Metabolic engineering involves, by necessity, renewable raw materials and is a vital component of the future development of renewable feedstocks for fuels and chemicals. 

The over-all process of crystal growth in a seeded solution is analogous to other mass transfer situations encountered in chemical engineering and may be treated as a diffusional step in series with a surface reaction step. Solution supersaturation provides the driving force required for each step, as portrayed schematically in Fig. 12. First, solute molecules or ions diffuse through the solution to the growing crystal. Second, upon reaching the surface, the molecules or ions must be accepted and incorporated into the crystal lattice. 

Gas liquid reactions may conform to various mechanisms. Under certain conditions, the absorption and reaction may conform to a slow" reaction mechanism. By this term, we mean that a gaseous species A is absorbed, diffuses through the film,.and tben reacts ip the bulk liquid. Thus, according to film theory, the processes of chemical reaction and diffusion become two steps in series for a slow reaction. The absorption rate in this case is almost unaffected by a chemical reaction. In the limiting case, where the concentration profile of the absorbing species in the liquid film is flat, the reaction is often called a very slow reaction, and the process of absorption is said to be in the kinetically-controlled regime ... 

This should not be surprising, since the activation and mass-transport processes always occur in series and should combine to determine the overall rate in the same way as several-activation controlled steps in series. 

For the curves of Fig. 27, it is shown (183) that the best agreement with experimentally observed peak maxima and areas for the second species is obtained by the model based on three or four steps in series. The values found for In give a straight line when plotted against 1/T, as required by the model, and the activation energy of the hydrogenation can be obtained from such a curve. After 30 s of reaction, only the first species is... 

When heterogeneous reactions are carried out at steady state, the rates of each of the three reaction steps in series (adsorption, surface reaction, and desorption) are equal to one another ... 

The analysis of Langmuir [/. Am. Chem. Soc. 40 1361 (1918)] and Hinshelwood (Kinetics of Chemical Change, Oxford, 1940) form the basis for the simplified treatment of kinetics on heterogeneous catalysts. For a solid catalyzed reaction between gas phase reactants A and B, the postulated mechanism consists of the following steps in series ... 

Multiphase reactions can be significantly affected by how well mixed the system is and how intimately dispersed the phases are. The reason for this is easy to explain, but more difficult to quantify although the course of any reaction is determined exclusively by the local concentrations of the reactants and the intrinsic reaction kinetic rates, in any real reactive system, the local reactant concentrations depend not only on how fast the reactants are depleted by the reaction, but also on how fast they are locally replenished from the bulk of the phases in which they initially reside. The latter phenomenon is directly related to the existence of a mass transfer step (in series with the reaction step), which determines the rate at which the reactants in different phases are brought in contact with each other. In many cases, especially if the rate of reaction is fast with respect to the mass transfer rate, the latter mechanism can become controlling over the former, and the overall reaction process is dominated by mass transfer and, hence, multiphase mixing. 

The same methodology is applicable to more complex reaction schemes involving combinations of steps in series and in parallel (60, 61). If the... 

With two electrocatalytic steps in series, the concentration of the intermediate B (Eq. 70) goes through a maximum with time (or space-time for a flow reactor). Solution of the kinetic equations for each species (60) yields for the simple case of first-order reactions... 

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