Two first-order reactions in series

According to Eq. (4.1.10) the selectivity of the intermediate product B is given by (cb,o = 0, constant volume, batch reactor)  

Sometimes two kinds of seledivities are considered, namely, the integral (global) selectivity S defined by Eq. (4.3.41) and the so-called differential (instantaneous) selectivity s, which is the ratio of the rate of the formation of a product to the rate of conversion of the reactant  

for consecutive reactions, the selectivity depends on the ratio of the rate constants as well as on the reaction progress and conversion of the reactant. 

This method to derive approximate solutions for kinetic equations is called the principle of quasi-stationarity (first introduced by Max Eodenstein, see box), and is very helpful for the kinetic evaluation of complex reaction systems as the mathematical treatment of the kinetic systems becomes simpler. (Note that for the example discussed above the Eodenstein principle is not needed, as the exact equations (4.3.38)-(4.3.40) can be derived quite easily this example was just chosen to show this principle.) 

Kinetics is not only important for chemical or biological processes, but also for everyday life, as inspected in Example 4.3.1 for the social process of the birth and death of a rumor with interacting subpopulations of ignorants, spreaders, and stiflers. 

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The kinetics observed are typical of two first-order reactions in series. 

Figure 7.3 also compares the evolution of the concentrations of the intermediate Q and the product in case of two first-order reactions in series in a CSTR with that in a batch or plug flow reactor. For constant density, the mass balance for the reaction components in a CSTR are ... 

The dehydrogenation of n-butenes, diluted with steam, on a Shell 205 catalyst can be described approximately by two first-order reactions in series (A B C) ... 

Figure 4.11 Two first-order reactions in series in a batch reactor,...

For consecutive reactions where the intermediate is the desired product, the local selectivity and the maximum yield are decreased because of the concentration differences between the bubble phase and the dense phase. Consider the simple case of two first-order reactions in series, and assume Model II is applicable ... 

For two first-order reactions in series (A- B C) taking place in a porous catalyst plate with thickness 2L [rate constants k , j and k 2 (m kg cat s )], the differential equations for the reactant A (feedstock) and the intermediate B are ... 

Influence of Reactor Type on Product Yields and Selectivity Let us consider two first-order reactions in series A—C. In batch and plug flow reactors, the yield of intermediate B is given by Eq. (4.3.39), derived in Section 4.3.2.1 ... 

It has been stated by Boudart that the steady-state approximation (SSA) can be considered as the most important general technique of applied chemical kinetics [9]. A formal proof of this hypothesis that is applicable to all reaction mechanisms is not available because the rate equations for complex systems are often impossible to solve analytically. However, the derivation for a simple reaction system of two first-order reactions in series demonstrates the principle very nicely and leads to the important general conclusion that, to a good approximation, the rate of change in the concentration of a reactive intermediate, X, is zero whenever such an intermediate is slowly formed and rapidly disappears. 

Problem 8-4 (Level 3) A pilot plant is being operated to test a new catalyst for the partial oxidation of naphthalene to phthalic anhydride. The chemistry of this process can be approximated as two first-order reactions in series ... 

If the two steps of first-order reactions in series have very different values for their rate constants, we can approximate the overall behavior as follows ... 

However, we can exclude to some extent the existence of different elementary mechanisms in the high or low pressure region. In fact, for two first order steps in series - mass transfer and chemical reaction - theintrinsic rate constant k can be easely calculated from the experimental one k xp and the... 

For a first-order reaction, the mass transfer coefficient can be combined with the effective rate constant to give an overall coefficient, Kq, since we have two first-order steps in series. 

When reactions in series are considered, it is not possible to draw any very satisfactory conclusions without working out the product distribution completely for each of the basic reactor types. The general case in which the reactions are of arbitrary order is more complex than for parallel reactions. Only the case of two first-order reactions will therefore be considered ... 

Reactor capacity per unit volume appears to depend on four resistances in series the gas-phase transfer resistance, two liquid-phase transfer resistances, and the kinetic resistance. The highest resistance limits the capacity of the reactor. The four resistances have the unit of time and each one individually represents the time constant of the particular process under study. For example, 1 lkjigl is the time constant for the transfer of A from the bulk of the gas through the gas film to the gas-liquid interface. The same holds for the three other resistances. For a first-order reaction in a batch reactor, for example, the concentration after a certain time is given by C/C0 = exp(-r/r), in which r = 1/ A is the reaction time constant. For processes in series the individual time constants can be added to find the overall time constant of the total process. 

This strong influence of the IL layer on selectivity cannot be explained by the influence of the IL on the effective concentrations of COD and COE compared to the uncoated catalyst For two first-order reactions occurring in series in a porous catalyst (here hydrogenation of COD to COE and to COA), the maximum yield of the intermediate COE in the absence of any mass transfer resistances (//po e = V... 

Mass transfer resistances lead to a lower effective rate compared to the intrinsic chemical reaction, but may also significantly change the selectivity of parallel and consecutive reactions. In the following, this is discussed for two first-order reactions occurring in series or parallel, for simplification, the influence of external mass transfer is only discussed for a non-porous catalyst (to exclude pore diffusion), and the effect of pore diffusion is examined for a negligible influence of external mass transfer. Other more complicated cases are treated elsewhere (Baerns et al, 2006 Levenspiel, 1999 froment and Bischoff, 1990). 

For two first-order reactions occurring in series on the external surface Am,ex (m kg cat) of non-porous catalyst (reaction A- B- C), the effective reaction... 

Consider two, first-order reactions occurring in series, i.e.,... 

In the slurry process, the hydrolysis is accompHshed using two stirred-tank reactors in series (266). Solutions of poly(vinyl acetate) and catalyst are continuously added to the first reactor, where 90% of the conversion occur, and then transferred to the second reactor to reach hiU conversion. Alkyl acetate and alcohols are continuously distilled off in order to drive the equiUbrium of the reaction. The resulting poly(vinyl alcohol) particles tend to be very fine, resulting in a dusty product. The process has been modified to yield a less dusty product through process changes (267,268) and the use of additives (269). Partially hydroly2ed products having a narrow hydrolysis distribution cannot be prepared by this method. 

Use the F(t) curve for two identical CSTR s in series and the segregated flow model to predict the conversion achieved for a first-order reaction with k = 0.4 ksec-1. The space time for an individual reactor is 0.9 ksec. Check your results using an analysis for two CSTR s in series. 

That is, /or a first-order reaction, the two stages must be of equal size to minimize V. The proof can be extended to an N-stage CSTR. For other orders of reaction, this result is approximately correct. The conclusion is that tanks in series should all be the same size, which accords with ease of fabrication. 

The optimum size ratio for two mixed flow reactors in series is found in general to be dependent on the kinetics of the reaction and on the conversion level. For the special case of first-order reactions equal-size reactors are best for reaction orders n > 1 the smaller reactor should come first for n < 1 the larger should come first (see Problem 6.3). However, Szepe and Levenspiel (1964) show that the advantage of the minimum size system over the equal-size system is quite small, only a few percent at most. Hence, overall economic consideration would nearly always recommend using equal-size units. 

Consider a two-step first-order irreversible reactions in series... 

Wu and Gschwend (1986) reviewed and evaluated several kinetic models to investigate sorption kinetics of hydrophobic organic substances on sediments and soils. They evaluated a first-order model (one-box) where the reaction is evaluated with one rate coefficient (k) as well as a two-site model (two-box) whereby there are two classes of sorbing sites, two chemical reactions in series, or a sorbent with easily accessible sites and difficultly accessible sites. Unfortunately, the latter model has three independent fitting parameters kx, the exchange rate from the solution to the first (accessible sites) box k2, the exchange rate from the first box to the... 

CSTRs in Series. A first-order reaction with no volume change (u = Uq) is to be carried out in two CSTRs placed in series (Figure 4-3). The effluent con-... 

Tanks-in-Series Model Versus Dispersion Model. We have seen that we can apply both of these one-parameter models to tubular reactors using the variance of the RTD. For first-order reactions the two models can be applied with equal ease. However, the tanks-in-series model is mathematically easier to use to obtain the effluent concentration and conversion for reaction orders other than one and for multiple reactions. However, we need to ask what would be the accuracy of using the tanks-in-series model over the dispersion model. These two models are equivalent when the Peclet-Bodenstein number is related to the number of tanks in series, n, by the equation ... 

Reactions in Series Two first-order elementary reactions in series are... 

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