Kinetics rate equations

The solution to the usual macroscopic kinetic rate equations for the reactant and product concentrations yields... 

The component mass balance equation, combined with the reactor energy balance equation and the kinetic rate equation, provide the basic model for the ideal plug-flow tubular reactor. 

The model is given by the following kinetic rate equation... 

Let us consider the determination of two parameters, the maximum reaction rate (rITOIX) and the saturation constant (Km) in an enzyme-catalyzed reaction following Michaelis-Menten kinetics. The Michaelis-Menten kinetic rate equation relates the reaction rate (r) to the substrate concentrations (S) by... 

Sharma et al. [153] have devised a gentle accelerated corrosion test using a kinetic rate equation to establish appropriate acceleration factors due to relative humidity and thermal effects. Using an estimate for the thermal activation energy of 0.6 eV and determining the amount of adsorbed water by a BET analysis on Au, Cu and Ni, they obtain an acceleration factor of 154 at 65°C/80% RH with respect to 25 °C/35-40% RH. 

Formation of products in paraffin cracking reactions over acidic zeolites can proceed via both unimolecular and bimolecular pathways [4], Based on the analysis of the kinetic rate equations it was suggested that the intrinsic acidity shows better correlation with the intrinsic rate constant (kinl) of the unimolecular hexane cracking than with the apparent rate constant (kapp= k K, where K is the constant of adsorption equilibrium). In... 

Catalysis by such a mechanism can be accounted for in a kinetic rate equation by including as a factor the catalyzing mineral s surface area. Sung and Morgan (1981), for example, in studying the oxidation on Mn11 by dissolved dioxygen,... 

The initial structure of the program is then followed by statements reflecting the dynamic model equations. These are also provided with comment lines with surrounding braces to distinguish them from the executable program lines. Note that the kinetic rate equations are expressed separately apart from the balance equations, to provide additional simplicity and additional flexibility. The kinetic rates are now additional variables in the simulation and the rates can... 

This section mainly builds upon classic biochemistry to define the essential building blocks of metabolic networks and to describe their interactions in terms of enzyme-kinetic rate equations. Following the rationale described in the previous section, the construction of a model is the organization of the individual rate equations into a coherent whole the dynamic system that describes the time-dependent behavior of each metabolite. We proceed according to the scheme suggested by Wiechert and Takors [97], namely, (i) to define the elementary units of the system (Section III. A) (ii) to characterize the connectivity and interactions between the units, as given by the stoichiometry and regulatory interactions (Sections in.B and II1.C) and (iii) to express each interaction quantitatively by... 

Aiming to construct explicit dynamic models, Eqs. (5) and (6) provide the basic relationships of all metabolic modeling. All current efforts to construct large-scale kinetic models are based on an specification of the elements of Eq (5), usually involving several rounds of iterative refinement For a schematic workflow, see again Fig. 4. In the following sections, we provide a brief summary of the properties of the stoichiometric matrix (Section III.B) and discuss the most common functional form of enzyme-kinetic rate equations (Section III.C). A selection of explicit kinetic models is provided in Table I. TABLE I Selected Examples of Explicit Kinetic Models of Metabolisin 1 ... 

Although we do not necessarily agree that the exact mechanism is always irrelevant, an approximative scheme to represent enzyme-kinetic rate equations indeed often allows to deduce putative properties of the network in a quick and straightforward way. Consequently, the utilization of approximative kinetics in the analysis of metabolic networks provides a reasonable strategy toward a large-scale dynamic view on cellular metabolism. In the following we give a brief summary of the most common approaches. 

To obtain the explicit functional form of the convenience kinetics rate equation, recall the Bi-Bi random-order mechanism already considered in Section III.C.5 and depicted again below ... 

Petersen [12] points out that this criterion is invalid for more complex chemical reactions whose rate is retarded by products. In such cases, the observed kinetic rate expression should be substituted into the material balance equation for the particular geometry of particle concerned. An asymptotic solution to the material balance equation then gives the correct form of the effectiveness factor. The results indicate that the inequality (23) is applicable only at high partial pressures of product. For low partial pressures of product (often the condition in an experimental differential tubular reactor), the criterion will depend on the magnitude of the constants in the kinetic rate equation. 

Though there are many problems related to solid catalysts, we consider only those which are related to the development of kinetic rate equations needed in design. We simply assume that we have a catalyst available to promote a specific reaction. We wish to evaluate the kinetic behavior of reactants in the presence of this material and then use this information for design. 

In a graphite atomizer, the atoms will appear according to a kinetic rate equation which will probably contain an exponential function. As the number of atoms in the atom cell increases, so does the rate of removal, until, at the absorption maximum (peak height measurement), the rate of formation equals the rate of removal. Thereafter, removal dominates. 

Information on the steps in a reaction mechanism can be extended significantly by isotopic tracer measurements, especially by transient tracing [see Happel et al. (54,55)]. Studies by Temkin and Horiuti previously referenced here have been confined to steady-state isotopic transfer techniques. Modeling with transient isotope data is often more useful since it enables direct determination of concentrations of intermediates as well as elementary step velocities. When kinetic rate equations alone are used for modeling, determination of these parameters is more indirect. 

To applying Eq. (2.47) to non-isothermal problems, it is necessary to generalize it by introducing temperature-dependent constants. The basic approach was proposed by Ziabicki94,95 who developed a quasi-static model of non-isothermal crystallization in the form of a kinetic rate equation ... 

To investigate the impact of kinetics on reactor design, we have to (i) develop the pertinent kinetic rate equation (ii) insert this rate law into the equation for the reactor we intend to operate and (iii) integrate over all degrees of conversion. 

To investigate the impact of inhibition on reactor design, the procedure again is as follows (i) we have to develop the pertinent kinetic rate equation, (ii) insert this rate law into the equation for the reactor we intend to operate, and (iii) integrate over all applicable degrees of conversion. As the two most frequent cases of inhibition are the occurrence of substrate and (even more often) product inhibition, we will treat those two cases in the following section. We will just mention the equation for substrate inhibition, but we will develop step-by-step the equation for product inhibition. 

For carrier-mediated transport, the rate of movement across a membrane will now be constant, since flux is dependent on the capacity of the membrane carriers and not the mass of the chemical to be transported. These processes are described by zero-order kinetic rate equations of the form ... 

A critical component to any mathematical model of pyrolysis, ignition, or even flame spread is the modeling of small-scale thermal degradation. Traditionally, thermal degradation processes in solids are considered to be analogous to chemical reactions in gases and liquids and are modeled in terms of sets of kinetic rate equations, typically of the form... 

There are p + 4 unknowns however, there are 4 flow equations as listed above arid xv element conservation equations. Just as in the solution of the equilibrium flame temperature problem discussed in section II. B. 5., M - a additional equations are required. Except instead of using the equilibrium equations, one must adopt the chemical kinetic rate equations. The form used with the present problem is ... 

The technique for coupling the chemical kinetic rate equations to the combustion process taking place in a rocket combustion chamber has not been devised. A detailed solution of the combustion chamber kinetics problem requires combination of the relations governing mixing, droplet burning, chemical reaction rates and combustion chamber flow characteristics. It is neither obvious that the complete solution to the complex combustion kinetics problem is possible nor that the efforts in this direction are wisely undertaken on the basis of present understanding of the more fundamental processes. 

Kinetic Rate Equation and Heat Transfer Coefficients. 

Kinetic rate equations of different complexity with 2 to 8 parameters were tested for the simulation of the reactor behaviour. Finally, the semi-empirical three parameter rate equation 20) was chosen for the simulation because rate expressions of higher complexity yielded no better simulation of the reactor behaviour and showed larger correlations between the estimated parameters in the given ranges of the process variables. 

The first attempt to relate changing dielectric properties to kinetic rate equations was by Kagan et al.S9), working with a series of anyhydride-cured epoxies. Building on Warfield s assumed correlation between d log (g)/dt and da/dt, where a is the extent of epoxide conversion, they assumed a proportionality between a and log (q), and modeled the reaction kinetics using the equation... 

In 1973, Leonov et al. 61) introduced a kinetic rate equation for the low-pressure copper-zinc oxide-alumina catalyst for temperatures between 220 and 260°C. The rate equation was proposed to be... 

General Kinetic Rate Equations. The rate of a bi-molecular reaction is given by... 

Kinetic rate equations may change per batch of catalyst manufactured and may vary between different lots. 

From the above it is clear that kinetic rate equations are afflicted with uncertainties, which can be caused by inaccurate data and the fitting procedures used, and also by the experimental methods and the differences in catalyst material and its quality. It is not possible to conclude from the fitted rate equations, what the reaction mechanism of the catalysis is. 

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