Steady state parallel reactions

The laboratory studies utilized small-scale (1-5-L) reactors. These are satisfactoiy because the reaction rates observed are independent of reac tor size. Several reac tors are operated in parallel on the waste, each at a different BSRT When steady state is reached after several weeks, data on the biomass level (X) in the system and the untreated waste level in the effluent (usually in terms of BOD or COD) are collected. These data can be plotted for equation forms that will yield linear plots on rec tangular coordinates. From the intercepts and the slope or the hnes, it is possible to determine values of the four pseudo constants. Table 25-42 presents some available data from the literature on these pseudo constants. Figure 25-53 illustrates the procedure for their determination from the laboratory studies discussed previously. 

Suppose there are two parallel, first-order reactions in a steady-state CSTR. Show that neither the fed-batch nor fast-fill-and-hold strategies can achieve a bumpless startup if the reactions have different rate... 

Two fixed-bed reactors can be used in parallel, one reacting and the other regenerating. However, there are many disadvantages in carrying out this type of reaction in a packed bed. The operation is not under steady state conditions, and this can present control problems. Eventually, the bed must be taken off line to replace the solid. Fluidized beds (to be discussed later) are usually preferred for gas-solid noncatalytic reactions. 

The MS analysis shows that the C02 profile led that of the PO profile (results not shown). The step switch results further confirm that C02 formation is faster than PO formation and that both reactions take place in parallel. GC analysis of the steady state effluent stream from the reactor revealed that propylene conversion was 10.5% at 250 °C product formation rates were determined to be 1.33, 0.12, and 34.3 pmol/min, respectively, for acetone, PO,... 

A liquid-phase reaction takes place in two CSTRs operating (at steady-state) in parallel at... 

However, we have to reflect on one of our model assumptions (Table 5.1). It is certainly not justified to assume a completely uniform oxide surface. The dissolution is favored at a few localized (active) sites where the reactions have lower activation energy. The overall reaction rate is the sum of the rates of the various types of sites. The reactions occurring at differently active sites are parallel reaction steps occurring at different rates (Table 5.1). In parallel reactions the fast reaction is rate determining. We can assume that the ratio (mol fraction, %a) of active sites to total (active plus less active) sites remains constant during the dissolution that is the active sites are continuously regenerated after AI(III) detachment and thus steady state conditions are maintained, i.e., a mean field rate law can generalize the dissolution rate. The reaction constant k in Eq. (5.9) includes %a, which is a function of the particular material used (see remark 4 in Table 5.1). In the activated complex theory the surface complex is the precursor of the activated complex (Fig. 5.4) and is in local equilibrium with it. The detachment corresponds to the desorption of the activated surface complex. 

A procedure used to assist in identifying sequential mechanisms when the double-reciprocal plots exhibit parallel lines ". In some cases, bireactant mechanism can have various collections of rate constants that result in so-called parallel line kinetics, even though the mechanism is not ping pong. However, if the concentrations of A and B are kept in constant ratio with respect to each other, a sequential mechanism in a 1/v v. 1/[A] plot would be nonlinear (since in the denominator the last term of the double-reciprocal form of the rate expression contains [A] for example, for the steady-state ordered Bi Bi reaction scheme in which [B] = a[A], the double-reciprocal rate expression becomes 1/v =... 

This linearization of the tight-binding scheme allows the investigator the opportunity to calculate values for [Etotai] and Ki, the dissociation constant for the inhibitor. In the Henderson plot, [Itotai]/(l v/Vo) is plotted as a function of vjv where Vq is the steady-state velocity of the reaction in the absence of the inhibitor. The slope of the line is the apparent dissociation constant for the inhibitor. Secondary plots (from repeating the inhibition experiment at different substrate concentrations) will yield the Ki value. The vertical intercept is equal to [Etotai]- Hence, repeating the experiment at a different concentration of enzyme will produce a parallel line. 

In treating parallel reaction, two concepts are often used (i) the concept of rate-determining path, in which the fastest path is the rate-determining path, and (ii) the concept of steady state, also called the concept of quasi-stationary states of trace-level intermediates. 

Eventually, in steady state, 0H will be large (approaching 0.9, say). Most of the surface becomes occupied by the electrochemical desorption reaction. The discharge reaction into the remaining small area of the bare metal of the electrode will occur in parallel to that of the electrochemical desorption reaction (at steady state), which will have the same rate in the desorption reaction. Both reactions occur in parallel. 

Chapter Njjgludes examples of parallel reactions, e.g., the attack of various nucleophiles on acylchymotrypsins, measured by steady state and pre-steady state kinetics. 

The parametric approach, which is not strictly needed for a single Gray-Scott reaction, works very well for an arbitrary number of parallel reactions and for continuous mixtures. Figure 16 shows a case of two parallel reactions for which an isola and a mushroom coexist. Because the notions of continuous mixtures and reactions will be treated in Chapter 8, G H and in the group of papers listed in the Index of Subjects in Publications under the heading Continuous mixtures, we can be very brief and start with the nondimensional equations. Let x be the index of the mixture whose species are /4(x). The steady-state concentration of the material with index in (x, x + dx) is V(x)dx, the feed concentration a(x)dx and the conversion U(x) = 1 - V/(x)/a(x), the last being defined only for values of x for which a(x) is not zero. B, the autocatalytic agent, forms itself as an undifferentiated product whose concentration is W. The rate of the first reaction, and hence p,(x), depends on the... 

The non-dimensionalization used in this work is perhaps the simplest, but it suffers from the defect that important physical bifurcation parameters are not isolated. The simple cuspoid diagrams are probably not those that would be obtained from experiments, where the residence time is a convenient parameter. Balakotaiah and Luss (1983) considered such a formulation for two parallel or simultaneous reactions the diagrams for the case of sequential reactions are similar, at least when the activation energies are equal. The maximum multiplicity question, however, is independent of the formulation and we conjecture that diagrams with seven steady states could be found in a small region of parameter space, though we have not looked for them. 

Balakotaiah, V. and Luss, D., 1982b, Exact steady-state multiplicity criteria for two consecutive or parallel reactions in lumped-parameter-systems. Chem. Engng ScL 37,433-445. 

In summary, Ogwada and Sparks (1986c) developed a model and assumed that the adsorption of ions from solution by soil particles occurs in a series rather than a parallel reaction mode. Thus, mass-transfer processes and CR occur consecutively. Under the steady-state approximation, the rate of mass transfer is approximately equal to the rate of the reaction, so that instantaneous change in the concentration of CA with time approaches... 

The lag between density cell response and reactor events were considerably less for this example and the figures ignore any correction. After establishing a "steady state" response to the monomer feed (about 160 minutes into the reaction), the incremental increase of the feed rate is seen not to alter the overall fractional conversion since the rate of polymerization increases to parallel the monomer feed rate. At the end of this set of data the rate is 2-3 times that observed earlier before the feed. 

If a chemical reaction is operated in a flow reactor under fixed external conditions (temperature, partial pressures, flow rate etc.), usually also a steady-state (i.e., time-independent) rate of reaction will result. Quite frequently, however, a different response may result The rate varies more or less periodically with time. Oscillatory kinetics have been reported for quite different types of reactions, such as with the famous Belousov-Zha-botinsky reaction in homogeneous solutions (/) or with a series of electrochemical reactions (2). In heterogeneous catalysis, phenomena of this type were observed for the first time about 20 years ago by Wicke and coworkers (3, 4) with the oxidation of carbon monoxide at supported platinum catalysts, and have since then been investigated quite extensively with various reactions and catalysts (5-7). Parallel to these experimental studies, a number of mathematical models were also developed these were intended to describe the kinetics of the underlying elementary processes and their solutions revealed indeed quite often oscillatory behavior. In view of the fact that these models usually consist of a set of coupled nonlinear differential equations, this result is, however, by no means surprising, as will become evident later, and in particular it cannot be considered as a proof for the assumed underlying reaction mechanism. 

---