Nonisothermal reaction kinetics

Except for radioactive decays, other reaction rate coefficients depend on temperature. Hence, for nonisothermal reaction with temperature history of T(t), the reaction rate coefficient is a function of time k(T(t)) = k(t). The concentration evolution as a function of time would differ from that of isothermal reactions. For unidirectional elementary reactions, it is not difficult to find how the concentration would evolve with time as long as the temperature history and hence the function of k(t) is known. To illustrate the method of treatment, use Reaction 2A C as an example. The reaction rate law is (Equation 1-51) 

The above equation is equivalent to Equation 1-51 by making a equivalent to kt. Hence, the solution can be obtained similar to Equation 1-53 as follows  

In general, for unidirectional elementary reactions, it is easy to handle non-isothermal reaction kinetics. The solutions listed in Table 1-2 for the concentration evolution of elementary reactions can be readily extended to nonisothermal reactions by replacing kt with a = j k df. The concepts of half-life and mean reaction time are not useful anymore for nonisothermal reactions. 

The most often encountered thermal history by geologists is continuous cooling from a high temperature to room temperature (such as cooling of volcanic rocks, plutonic rocks, and metamorphic rocks). One of the many ways to approximate the cooling history is as follows  

That is, k decreases with time exponentially with a timescale of x. Therefore, a can be found to be 

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For nonisothermal reaction kinetics, Ganguly (1982) applied the above solution for a small time interval At 0. In this time interval, Xq and kbi may be regarded as constant. Hence,... 

To obtain the nonisothermal reaction kinetics, the sulfuric acid-containing coke as heated at a constant rate of 5°C/min and the volume of evolving individual reaction products was monitored vs. the change in temperature. Under the conditions of this experiment the regeneration reaction starts around 200°C and is practically completed at 450°C as indicated by the evolution of the reaction products as shown in Figure 2. 

Knowing A and for a given reaction in the temperature interval of interest, one can calculate reaction kinetics in nonisothermal experiments. One can also, in reverse, derive values for the activation energy, the pre-exponential factor, and the order of the reaction fi-om nonisothermal experiments. For this purpose one inserts Eq. (7) into Eq. (3) of Fig. 2.8 and gets Eq. (8) as an expression for the nonisothermal reaction kinetics. For analysis, one may take the logarithm on both sides of the equation to make the exponential disappear. In addition, one may differentiate both sides with respect to ln[A], to get an explicit equation for the reaction order n. The result of this mathematical operation is shown in Eq. (9). This is a somewhat arduous equation, usually attributed to Freeman and Carroll. Note that experimentally one knows the parameters concentration, [A], rate, -d[A]/dt,... 

This development has been generalized. Results for zero- and second-order irreversible reactions are shown in Figure 10. Results are given elsewhere (48) for more complex kinetics, nonisothermal reactions, and particle shapes other than spheres. For nonspherical particles, the equivalent spherical radius, three times the particle volume/surface area, can be used for R to a good approximation. 

In general, various types of traps and recombination centers may be present, and their involvement in the reaction kinetic process will greatly change with temperature. The temperature range in which a specific range dominates must, therefore, be determined. This is most conveniently achieved with the aid of nonisothermal temperature scans, during which TSL and TSC are monitored. Of course, the microscopic physical and chemical nature of traps cannot be determined with these methods. 

Those simplified models are often used together with simplified overall reaction rate expressions, in order to obtain analytical solutions for concentrations of reactants and products. However, it is possible to include more complex reaction kinetics if numerical solutions are allowed for. At the same time, it is possible to assume that the temperature is controlled by means of a properly designed device thus, not only adiabatic but isothermal or nonisothermal operations as well can be assumed and analyzed. 

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... 

Nonisothermal reactions Numerous kinetic investigations of the thermal reactions of solids have used rising temperature techniques, often during a linear rate of reactant temperature increase. The kinetic analysis then requires the solution of three equations ... 

Mass transfer in combination with even quite "normal" reaction kinetics can produce a wealth of phenomena including multiple steady states, instabilities, and oscillations. An example is the behavior of nonisothermal catalyst particles outlined in Section 9.5.2. Such phenomena are covered in detail in standard texts on reaction engineering, to which the reader is referred. The examination in this section will remain restricted to effects produced by vagaries of multistep or multiple simultaneous reactions. 

In another investigation, Gallagher and Johnson (147) compared isothermal and nonisothermal methods to study the reaction kinetics of the thermal decomposition of CaC03. According to isothermal kinetics studies, the reaction... 

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... 

The analysis of activity and selectivity in nonisothermal reactions, of course, requires the numerical solution to the mass- and energy-balance equations with the boundary conditions of equation (7-47). In nondimensional form, and for first-order kinetics, these are... 

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. 

In some simple cases of reaction kinetics, it is possible to solve the balance equations of the ideal, homogeneous reactors analytically. There is, however, a precondition isother-micity if nonisothermal conditions prevail, analytical solutions become impossible or, at least, extremely difficult to handle, since the energy and molar balances are interconnected through the exponential temperature dependencies of the rate and equilibrium constants (Sections 2.2 and 2.3). Analytical solutions are introduced in-depth in the literature dealing... 

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. 

Consider the case of a nonisothermal reaction A B occurring in the interior of a spherical catalyst pellet of radius R (Figure 6.4). We wish to compute the effect of internal heat and mass transfer resistance upon the reaction rate and the concentration and temperature profiles within the pellet. If Z)a is the effective binary diffusivity of A within the pellet, and we have first-order kinetics, the concentration profile CA(f) is governed by the mole balance... 

Chapter 1 treated single, elementary reactions in ideal reactors. Chapter 2 broadens the kinetics to include multiple and nonelementary reactions. Attention is restricted to batch reactors, but the method for formulating the kinetics of complex reactions will also be used for the flow reactors of Chapters 3 and 4 and for the nonisothermal reactors of Chapter 5. 

Most kinetic experiments are run in batch reactors for the simple reason that they are the easiest reactor to operate on a small, laboratory scale. Piston flow reactors are essentially equivalent and are implicitly included in the present treatment. This treatment is confined to constant-density, isothermal reactions, with nonisothermal and other more complicated cases being treated in Section 7.1.4. The batch equation for component A is... 

The reactor feed mixture was "prepared so as to contain less than 17% ethylene (remainder hydrogen) so that the change in total moles within the catalyst pore structure would be small. This reduced the variation in total pressure and its effect on the reaction rate, so as to permit comparison of experiment results with theoretical predictions [e.g., those of Weisz and Hicks (61)]. Since the numerical solutions to the nonisothermal catalyst problem also presumed first-order kinetics, they determined the Thiele modulus by forcing the observed rate to fit this form even though they recognized that a Hougen-Watson type rate expression would have been more appropriate. Hence their Thiele modulus was defined as... 

There are several factors that may be invoked to explain the discrepancy between predicted and measured results, but the discrepancy highlights the necessity for good pilot plant scale data to properly design these types of reactors. Obviously, the reaction does not involve simple first-order kinetics or equimolal counterdiffusion. The fact that the catalyst activity varies significantly with time on-stream and some carbon deposition is observed indicates that perhaps the coke residues within the catalyst may have effects like those to be discussed in Section 12.3.3. Consult the original article for further discussion of the nonisothermal catalyst pellet problem. 

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. 

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