Reaction rates basic rate equation

Choosing a reactor for a given reaction is based on several considerations and combines reaction analysis with reactor analysis. Thus, we consider in this chapter the following aspects of reactions and reactors, much of which should serve as an introduction to chemists and a refresher to chemical engineers reaction rates, stoichiometry, rate equations, and the basic reactor types. 

Reaction Special condition Basic rate equation Driving force Adsorption term... 

The treatment given here follows that of Freeman and Carroll [531], which was also considered by others [534,569]. Again, the logarithmic form of the basic rate equation is used and the reaction order expressed in the form f(a) = fc(l — a)" so that, for incremental differences in (da/dT), (1 — a) and T 1, one can write... 

Again we consider the basic rate equation (4.1). As described in equation (4.2), the reaction rate column vector f is usually written in the form... 

When only one reaction is taking place, the only variables to bd considered are time and conversion. The reaction A —> R is the simplest of this type. It represents a molecular rearrangement. The basic rate equation for this reaction... 

These conditions are expressed in the basic rate equations. Consider, for example, the elementary reaction ... 

A kinetic analysis based on the Coats-Redfern method applied nonisothermal TGA data to evaluate the stability of the polymer during the degradation experiment. Of the different methods, the Coats-Redfern method has been shown to offer the most precise results because gives a linear fitting for the kinetic model function [97]. This method is the most frequent in the estimation of the kinetic function. It is based on assumptions that only one reaction mechanism operates at a time, that the calculated E value relates specifically to this mechanism and that the rate of degradation, can be expressed as the basic rate equation (Eq. 5.3). This method is an integral method that assumes various... 

The development of combustion theory has led to the appearance of several specialized asymptotic concepts and mathematical methods. An extremely strong temperature dependence for the reaction rate is typical of the theory. This makes direct numerical solution of the equations difficult but at the same time accurate. The basic concept of combustion theory, the idea of a flame moving at a constant velocity independent of the ignition conditions and determined solely by the properties and state of the fuel mixture, is the product of the asymptotic approach (18,19). Theoretical understanding of turbulent combustion involves combining the theory of turbulence and the kinetics of chemical reactions (19—23). 

The water elimination reactions of Co3(P04)2 8 H20 [838], zirconium phosphate [839] and both acid and basic gallium phosphates [840] are too complicated to make kinetic studies of more than empirical value. The decomposition of the double salt, Na3NiP3O10 12 H20 has been shown [593] to obey a composite rate equation comprised of two processes, one purely chemical and the other involving diffusion control, for which E = 38 and 49 kJ mole-1, respectively. There has been a thermodynamic study of CeP04 vaporization [841]. Decomposition of metal phosphites [842] involves oxidation and anion reorganization. 

A soluble gas is absorbed into a liquid with which it undergoes a second-order irreversible reaction. The process reaches a steady-state with the surface concentration of reacting material remaining constant at (.2ij and the depth of penetration of the reactant being small compared with the depth of liquid which can be regarded as infinite in extent. Derive the basic differential equation for the process and from this derive an expression for the concentration and mass transfer rate (moles per unit area and unit time) as a function of depth below the surface. Assume that mass transfer is by molecular diffusion. 

The starting point for the development of the basic design equation for a well-stirred batch reactor is a material balance involving one of the species participating in the chemical reaction. For convenience we will denote this species as A and we will let (— rA) represent the rate of disappearance of this species by reaction. For a well-stirred reactor the reaction mixture will be uniform throughout the effective reactor volume, and the material balance may thus be written over the entire contents of the reactor. For a batch reactor equation 8.0.1 becomes... 

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... 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

The units of the specific rate depend on the order of a power law rate equation and on the units of the rate of reaction which are basically, (mass)/(time) (volume). 

The obvious advantage is that the steady-state solution of an S-system model is accessible analytically. However, while the drastic reduction of complexity can be formally justified by a (logarithmic) expansion of the rate equation, it forsakes the interpretability of the involved parameters. The utilization of basic biochemical interrelations, such as an interpretation of fluxes in terms of a nullspace matrix is no longer possible. Rather, an incorporation of flux-balance constraints would result in complicated and unintuitive dependencies among the kinetic parameters. Furthermore, it must be emphasized that an S-system model does not necessarily result in a reduced number of reactions. Quite on the contrary, the number of reactions r = 2m usually exceeds the value found in typical metabolic networks. 

This equation takes into account the influence of the polarization f(n) and polarity f(e) of the solvents determining their ability to non-specific solvation and also their basicities B [22] which are accordingly to Koppell-Paim s quantitatively equal to OW-group displacement absorption band in / -spectrum of the phenol dissolved in given solvent, and electrophilicity accordingly to Reichardt ET characterizing their ability to introduce into acid-base interactions (specific solvation). Appropriateness of this equation for the generalization of experimental data of the dependencies of reactions rates (and also spectral characteristics of dissolved substances) on physical-chemical characteristics of the solvents has been proved by a number of hundred examples. 

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