Reaction from Nonisothermal Kinetics

On the basis of many experiments and theoretical considerations, Arnold et al. (158) showed that TG curves are strongly influenced by the experimental conditions, and hence the kinetic parameters calculated from these curves are fictitious and their determination is uncertain. The Arrhenius equation, taken from homogeneous kinetics, cannot be applied to nonisothermal heterogeneous reactions since the conditions of the equation are not fulfilled. 

In DSC, the extent of reaction is obtained by first determining the total area of the peak that corresponds to the complete reaction. The required information from the experiment is the fraction of the complete reaction at a series of temperatures so that nonisothermal kinetics procedures can be applied. At a specific temperature, the partial peak area is determined, and the fraction of the complete reaction at that temperature is determined by dividing the area of the peak up to that temperature by the total peak area. A typical endothermic peak in a DSC trace is shown in Figure 8.2. In this case, the temperatures at which a is to be determined are indicated as... 

The authors of [104] attempted a qualitative and even semiquantitative analysis of the thermokinetic interactions and the nonlinear nature of this highly exothermic system. The results obtained from nonisothermal modeling of the process are qualitatively consistent with the above experimental results. A detailed mathematical description of the kinetics and processes of heat and heat transfer in a CSTR made it possible to reproduce the appearance of temperature hysteresis and to demonstrate its dependence on the oxygen concentration, reaction time, and temperature of the reactor walls. Figure 8.6 shows the calculated self-heating... 

The chapter ends with a case study. Four different reduced kinetic models are derived from the detailed kinetic model of the phenol-formaldehyde reaction presented in the previous chapter, by lumping the components and the reactions. The best estimates of the relevant kinetic parameters (preexponential factors, activation energies, and heats of reaction) are computed by comparing those models with a wide set of simulated isothermal experimental data, obtained via the detailed model. Finally, the reduced models are validated and compared by using a different set of simulated nonisothermal data. 

Example 9.11 Modeling of a nonisothermal plug flow reactor Tubular reactors are not homogeneous, and may involve multiphase flows. These systems are called diffusion convection reaction systems. Consider the chemical reaction A -> bB described by a first-order kinetics with respect to the reactant A. For a nonisothermal plug flow reactor, modeling equations are derived from mass and energy balances... 

Kinetic Expressions. In this study, we have analyzed nonisothermal TGA data using the Chen-Nuttall equation, the widely accepted Coats-Redfern equation, and the Anthony-Howard equation. These equations are derived from simple rate expressions. The basic single reaction kinetic equation for the decomposition of a solid has been presented by Blazek (24) as... 

In work on the hydrogen evolution reaction at Hg from CF3S03 H30 (where the proton is present only as the unhydrated H30 ion) and from CF3SO3H in excess water (where the proton is present as the fully hydrated ion H9O4), Conway et directly derived the real entropies of activation for proton discharge at Hg by means of kinetic measurements at various temperatures employing a nonisothermal cell, i.e., with a reference electrode at... 

Since the kinetics of homogeneous and heterogeneous reactions are fundamentally different, Arnold et al. (157) have shown that the nonisothermal TG curve provides insufficient information for the purpose of reaction kinetic calculations. Mathematical considerations prove also that the parameters of the Arrhenius model cannot be calculated correctly from the TG curve by curve-fitting methods and that there is no unique correlation between the estimated parameters and the measured curves. Also, the correlation between A and D described as a compensation effect is certainly a mathematical... 

We mentioned earlier that it is possible to simplify somewhat the nonisothermal analysis by the use of a combined thermal parameter, /Fy, and reported the result presented by Liu in equation (7-28). Now, having examined the anatomy of the results presented in Figure 7.8, we can benefit from looking at these mapping functions in more detail. For our favorite Academic Reaction 1 (second-order kinetics can also be managed this way), we have the function given by equation (7-28) but now must add that certain regions of the 77 — 5 plot are excluded. The full story is... 

This equation may be solved by any of the several methods discussed in Chapter 6 (see Illustration 6.4), with the result as shown in Figure 7.9. Now, this is an isothermal calculation, but it shows both of the prominent features of the nonisothermal results of Figure 7.8, i.e., effectiveness > 1 for some ranges of 4>s multiplicity for high values of the parameter K. We have not said much in the text discussions to this point about negative-order kinetics this example should help us to keep in mind that reaction order < 0 is a world away from > 0. 

Nonisothermal operation of a liquid-phase CSTR with reversible exothermic nth-order chemical kinetics is the focus of this chapter. The reactor is well insulated from the surroundings, except for heat exchange across the cooling coil. The reaction scheme is... 

In a recent study Wang and Hofmann (1999) have stressed the importance of nonisothermal rate data. From a simple theoretical analysis they conclude that kinetic and transport data obtained under isothermal conditions in a laboratory reactor cannot logically be used to simulate any other type of reactor. This is because of the behavior of the Lipschitz constant L, which is a measure of the sensitivity of the reaction to different models. It tells us how any two models would diverge at the end of a reactor under different thermal conditions of operation. It is therefore a useful criterion for selecting the best model. It has been shown that L is different for different reactor models ... 

Simulation results are plotted as RX concentration versus time in Figure 19.12 for both isothermal and nonisothermal situations (it is only necessary to ignore the heat transfer groups to simulate the isothermal condition). Clearly, there is some effect at higher reaction times, but it is not significant. From the values of the Thiele modulus ((j> = 0.33), we can assume that the reaction is kinetically controlled. The low value of 0 also justifies the small effect of nonisothermicity. 

PM AX corresponds to the reaction rate constant, called maximal production rate in the present context, TEMP expresses a kind of dependence on time through temperature (a phenomenon common in nonisothermal reactions), it is called here temperature limitation, Iq is an external forcing function it is the global radiation on the water surface, U2 is a joint limitation factor describing light- and nutrient-dependence of primary production, or to use the language of reaction kinetics it expresses the deviation from mass-action type kinetics. 

The exciting issue of steady-state multiplicity has attracted the attention of many researchers. First the focus was on exothermic reactions in continuous stirred tanks, and later on catalyst pellets and dispersed flow reactors as well as on multiplicity originating from complex isothermal kinetics. Nonisothermal catalyst pellets can exhibit steady-state multiplicity for exothermic reactions, as was demonstrated by P.B. Weitz and J.S. Hicks in a classical paper in the Chemical Engineering Science in 1962. The topic of multiplicity and oscillations has been put forward by many researchers such as D. Luss, V. Balakotaiah, V. Hlavacek, M. Marek, M. Kubicek, and R. Schmitz. Bifurcation theory has proved to be very useful in the search for parametric domains where multiple steady states might appear. Moreover, steady-state multiplicity has been confirmed experimentally, one of the classical papers being that of A. Vejtassa and R.A. Schmitz in the AIChE Journal in 1970, where the multiple steady states of a CSTR with an exothermic reaction were elegantly illustrated. 

In isothermal operation, the temperatuie is maintained constant during the course of the reaction. This condition can be approached in practice by providing sufficient heat exchanger facilities to account for enthalpy effects arising during reaction. This mode of operation finds its major application in laboratory kinetic studies. Some energy in the form of heat is added to or removed from the reactor, but isothermal conditions are not satisfied. For adiabatic operation, the reactor is insulated to minimize heat transfer between the reactor contents and the surroundings. However, many industrial reactors attempt to operate in this manner. Most industrial reactors are described by nonisothermal operation. 

Values of at a particular temperature and pressure can be determined by measuring initiator concentration as a function of reaction time using a technique such as infrared spectroscopy. The experimental difficulty increases as half-life shortens, where special care must be taken to eliminate transient nonisothermal effects. Decomposition kinetics are summarized for a wide range of initiators in the Polymer Handbook [7] and in trade literature available from commercial suppliers. Of special note is the recent work of Buback and co-workers that systematically examines not only how Ea and AV vary with alkyl substituent for the peroxyester family (Scheme 4.2), but also how the substituent choice affects the decomposition pathway and initiator efficiency [1, 2]. 

Since PFR kinetics are essentially those of a BR (and such oscillations do not occur in BRs), no oscillations occur. The CSTRs, then, are used to grow the existing particles. As clearly demonstrated in Figure 17.5, by segregating particle nucleation from particle growth, oscillations are eliminated. Oscillations are also observed during nonisothermal solution polymerization in a CSTR, where interactions between the heat evolution and reaction rate cause instability. 

Time-dependent analyses are limited by the time constant of the equipment for applications of short time scale. Slow processes are, on the other hand, limited by the sensitivity of the equipment to register heat release over a longer time. In the latter case it is, however, frequently possible to establish the kinetics by discontinuous experimentation. The isothermal, slow process is driven to completion after a given time by heating or cooling and the isothermal kinetics is deduced from the residual reaction that occurred nonisothermally. This method is more time-consuming and needs a larger number of samples for complete analysis, but it has no limit in time scale, apart from the patience of the operator. 

Morales et al. [323] prepared bionanocomposites of PEA (derived from glycohc acid and 6-aminohexanoic add by in situ polymerization) reinforced with OMMTs. The most dispersed structure was obtained by addition of C25A organoclay. Evaluation of thermal stability and crystallization behavior of these samples showed significant differences between the neat polymer and its nanocomposite with C25A. Isothermal and nonisothermal calorimetric analyses of the polymerization reaction revealed that the kinetics was highly influenced by the presence of the silicate particles. Crystallization of the polymer was observed to occur when the process was isothermally conducted at temperatures lower than 145 °C. In this case, dynamic FTIR spectra and WAXD profiles obtained with synchrotron radiation were essential to study the polymerization kinetics. Clay particles seemed to reduce chain mobility and the Arrhenius preexponential factor. 

In fact, nonisothermal temperature cure is a different process from isothermal cure. The reaction kinetics, total reaction order, and even the reaction energy for epoxy systems may not be constant and same, but process dependent. Therefore, modifications need to be made to reflect the effects of such a difference. 

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