Second-order kinetics numerical results

Numerical solutions to equation 11.2.9 have been obtained for reaction orders other than unity. Figure 11.11 summarizes the results obtained by Levenspiel and Bischoff (18) for second-order kinetics. Like the chart for first-order kinetics, it is most appropriate for use when the dimensionless dispersion group is small. Fan and Bailie (19) have solved the equations for quarter-order, half-order, second-order, and third-order kinetics. Others have used perturbation methods to arrive at analogous results for the dispersion model (e.g. 20,21). 

This result has been corroborated by numerous other examples43,593,597,601,608-614 and fits well with the classical mechanism described in Scheme 2. Only a few examples of second-order reactions are known they concern in particular the alkaline hydrolysis of phosphonium salts 29s 86,615 and 30616. In both cases, the second-order kinetics are very likely the result of a direct substitution induced by the very high stability of the carbanion resulting from the P—C bond cleavage. 

The heat removal line is unchanged irrespective of the kinetics, and the fractional conversion XA has a qualitatively similar sigmoidal shape for second-order kinetics. Numerical values as in Example 6-11 have been put into both the mass balance and heat balance equations. Fractional conversion XA from both the mass balance and heat balance equations at effluent temperatures of 300, 325, 350, 375, 400, 425, 450, and 475 K, respectively, were determined using the Microsoft Excel Spreadsheet (Example6-12.xls). Table 6-8 gives the results of... 

First-order kinetics was chosen in writing Eq. (11-46), so that an analytical solution could be obtained. Numerical solutions for rj vs have been developed for many other forms of rate equations. Solutions include those for Langmuir-Hinshelwood equations with denominator terms, as derived in Chap. 9 [e.g., Eq. (9-32)]. To illustrate the extreme effects of reaction, Wheeler obtained solutions for zero- and second-order kinetics for a fiat plate of catalyst, and these results are also shown in Fig. 11-7. For many catalytic reactions the rate equation is approximately represented... 

Numerical solutions to equation (11.2.9) have been obtained for reaction orders other than unity. Figure 11.11 summarizes the results obtained by Levenspiel and Bischoff (18) for second-order kinetics. Like the chart for... 

For multisubstrate enzymatic reactions, the rate equation can be expressed with respect to each substrate as an m function, where n and m are the highest order of the substrate for the numerator and denominator terms respectively (Bardsley and Childs, 1975). Thus the forward rate equation for the random bi bi derived according to the quasi-equilibrium assumption is a 1 1 function in both A and B (i.e., first order in both A and B). However, the rate equation for the random bi bi based on the steady-state assumption yields a 2 2 function (i.e., second order in both A and B). The 2 2 function rate equation results in nonlinear kinetics that should be differentiated from other nonlinear kinetics such as allosteric/cooperative kinetics (Chapter 6, Bardsley and Waight, 1978) and formation of the abortive substrate complex (Dalziel and Dickinson, 1966 Tsai, 1978). 

The mechanisms and resulting kinetic equations are shown in Figure 4. Other mechanisms are possible as well as modifications of these—e.g., disproportion termination of chain reactions, and condensation between unlike monomers. The left sides of the equations represent the reactor operator (note that all resulting differential equations are nonlinear because of the second-order propagation and termination reactions). To this is added the complexity of considering separate equations for the thousands of separate species frequently required to define completely commercially useful polymers. Solution by direct application of classical techniques is impractical or impossible in most cases even direct numerical solution is often difficult. Simplifying assumptions or special mathematical techniques must be used (described below in the calculations of MWD). 

Reference was made earlier to some specific numerical studies reported for the axial dispersion model employing rate expressions other than first order. Some results given by Fan and Balie for half-, second-, and third-order kinetics are illustrated in Figure 5.21. In these results the parameter R is defined as... 

NUMERICAL RESULTS FOR SECOND-ORDER IRREVERSIBLE CHEMICAL KINETICS... 

The numerical procedures used to deal with experimental data are illustrated with results from an experiment in chemical kinetics recorded in table C.l. The reaction is first order but that will be ignored in the analysis of the data. Note that the data points have been obtained for equal increments in the independent variable time. It turns out that numerical techniques are especially easy to apply when this is the case. The second feature of this data set is that the precision of the time data is much higher than that of the concentration data. 

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