Series reactions, first-order

More terms of the series are usually not justifiable because the higher moments cannot be evaluated with sufficient accuracy from e)meri-mental data. A comparison of the fourth-order GC with other distributions is shown in Fig. 23-12, along with calculated segregated conversions of a first-order reaction. In this case, the GC is the best fit to the original. At large variances the finite value of the ordinate at... 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

First-order kinetics. Consider a first-order reaction studied with a series of initial concentrations, as depicted here. Show that the initial rate tangents come to a common intercept t" on the x axis, and find an expression for what time this is. 

Numerical calculations are the easiest way to determine the performance of CSTRs in series. Simply analyze them one at a time, beginning at the inlet. However, there is a neat analytical solution for the special case of first-order reactions. The outlet concentration from the nth reactor in the series of CSTRs is... 

A solution to Equation (8.12) together with its boundary conditions gives a r, z) at every point in the reactor. An analytical solution is possible for the special case of a first-order reaction, but the resulting infinite series is cumbersome to evaluate. In practice, numerical methods are necessary. 

For a first-order reaction n=l, and the dimensionless time is given by k t. Make a series of runs with different initial concentrations and compare the results, plotting the variables in both dimensional and dimensionless terms. 

Figure 5.33. Tanks-in-series with first-order reaction.

Tanks-in-series reactor configurations provide a means of approaching the conversion of a tubular reactor. In modelling, they are employed for describing axial mixing in non-ideal tubular reactors. Residence time distributions, as measured by tracers, can be used to characterise reactors, to establish models and to calculate conversions for first-order reactions. 

Referring back to the rate equation for a first-order reaction (Equation A1.2), we have a differential equation for which the derivative of the variable ([S]) is proportional to the variable itself. Such a system can be described by an infinite series with respect to time ... 

Schematic representation of the time dependence of the concentration of the first intermediate in a series of first-order reactions. Initial intermediate concentration is nonzero.

For first-order and pseudo first-order reactions of the series type several methods exist for determining ratios of rate constants. We will consider a quick estimation technique and then describe a more accurate method for handling systems whose kinetics are represented by equation 5.3.2. 

Series First-Order Reactions. Time-Percentage Reaction Relations for Various Relative Rate Constants... 

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

It is readily apparent that equation 8.3.21 reduces to the basic design equation (equation 8.3.7) when steady-state conditions prevail. Under the presumptions that CA in undergoes a step change at time zero and that the system is isothermal, equation 8.3.21 has been solved for various reaction rate expressions. In the case of first-order reactions, solutions are available for both multiple identical CSTR s in series and individual CSTR s (12). In the case of a first-order irreversible reaction in a single CSTR, equation 8.3.21 becomes... 

Comparison of performance of a series of N equal-size CSTR reactors with a plug flow reactor for the first-order reaction... 

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. 

Consecutive Reactions Where an Inter-mediate Is the Desired Product. Consecutive reactions in which an intermediate species (V) is the desired product are often represented as a series of pseudo first-order reactions... 

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. 

Extension of the Kunii-Levenspiel bubbling-bed model for first-order reactions to complex systems is of practical significance, since most of the processes conducted in fluidized-bed reactors involve such systems. Thus, the yield or selectivity to a desired product is a primary design issue which should be considered. As described in Chapter 5, reactions may occur in series or parallel, or a combination of both. Specific examples include the production of acrylonitrile from propylene, in which other nitriles may be formed, oxidation of butadiene and butene to produce maleic anhydride and other oxidation products, and the production of phthalic anhydride from naphthalene, in which phthalic anhydride may undergo further oxidation. 

Fitting of the boundary conditions requires the complete solution to be an infinite series. Numerical solutions may be preferable, and are always required for other than first order reactions. 

This program is designed to simulate the resulting residence time distributions based on a cascade of 1 to N tanks-in-series. Also, simulations with nth-order reaction can be run and the steady-state conversion obtained. A pulse input disturbance of tracer is programmed here, as in example CSTRPULSE, to obtain the residence time distribution E curve and from this the conversion for first order reaction. 

Recalling that for a first-order reaction it holds that /J/2 = 0.693/A , one can determine the half-life of the electrogenerated species Red from the value of k. In order to obtain a reliable value of kf it is always wise to have a series of measurements at different scan rates and possibly at different E values. 

Equations (5.16) of Table 5.1 refer to series first-order reactions. Of interest for the solvent extraction kinetics is a special case arising when the concentration of the intermediate, [Y], may be considered essentially constant (i.e., d[Y]/dt = 0). This approximation, called the stationary state or steady-state approximation, is particularly good when the intermediate is very reactive and present at very small concentrations. This situation is often met when the intermediate [Y] is an interfacially adsorbed species. One then obtains... 

These are called series first-order reactions, and they are commonly analyzed using numerical integration, as shown in Fig. 4. 

Laplace transformation is particularly useful in pharmacokinetics where a number of series first-order reactions are used to model the kinetics of drug absorption, distribution, metabolism, and excretion. Likewise, the relaxation kinetics of certain multistep chemical and physical processes are well suited for the use of Laplace transforms. 

A metabolite, molecular entity, or some other event/ process that precedes another component in a longer sequence of events or conversions. For example, the isoprenoid metabolite squalene is a precursor of cholesterol and glucose 6-phosphate is a precursor of glycogen, ribose, and pyruvate. See Series First Order Reaction Pulse-Chase Experiments... 

The simplest set of series reactions involves two first-order reactions ... 

SERIES FIRST ORDER REACTION PULSE CHASE EXPERIMENTS Predictor-corrector algorithm,... 

Fig. 12. General design chart for the tanks-in-series model described by eqn. (43), first-order reaction A -r r with no change in volume (e = 0). Ordinate gives the total volume of all the tanks in series divided by the volume of an ideal PFR which achieves the same conversion.-------, Constant kr -------, constant N.

CSTRs in series. The latter is often normalised by dividing by the volume of an ideal PFR required to perform the same duty. Different charts are required for each reaction rate expression. Figure 12 refers specifically to first-order kinetics, but other charts are available in, for instance refs. 17, 18 and 26. Figure 12 re-emphasises many of the points we have made already. In particular, the performance of the N CSTRs in series tends to that of a PFR of the same total volume as N becomes large and the PFR volume required to achieve a certain conversion for a first-order reaction is always smaller than the total volume of any array of CSTRs which perform the same duty. Charts in the form of Fig. 12 are particularly useful when performing approximate design calculations. 

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