Kinetic parameters from continuous Kinetics, chemical

While the decomposition of silacyclobutanes as a source of silenes has continued to be studied in the last two decades, the interest has largely focused on mechanisms and kinetic parameters. However, a few reports are listed in Table I of the presumed formation of silenes having previously unpublished substitution patterns, prepared either thermally or photo-chemically from four-membered ring compounds containing silicon. Two cases of particular interest involve the apparent formation of bis-silenes. Very low-pressure pyrolysis of l,4-bis(l-methyl-l-silacyclobutyl)ben-zene94 apparently formed the bis-silene 1, as shown in Eq. (2), which formed a high-molecular-weight polymer under conditions of chemical vapor deposition. 

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

The UV radiation disinfects germs in an aqueous system, which can be operated as plug flow, continuous flow, or other modes. The killing efficiency is controlled by many factors, which can be classified into two aspects disinfection kinetics and flow dynamics. Like many other processes in both chemical and environmental engineering, the mathematical modeling of the UV disinfection can be started from simulation of distribution of flow velocity together with definition of suitable kinetic model(s). The disinfection effect in terms of survival of pathogens as a function of operational conditions such as time and dose can then be estimated. Since the mathematical models involve many unknown parameters that must be experimentally determined, they are mainly... 

To study the behavior of the singular points in the vicinity of the MTBE vertex, Thiel et al. [8] used a continuation method with the Damkbhler number as continuation parameter. The results computed at p = 0.8 MPa are shown in Fig. 5.17. It can be observed that a stable node branch beginning from pure MTBE in the absence of chemical reaction moves away from MTBE vertex with rising Da. As the Damkohler number Da = 1.49 X 10 is reached, the stable node branch turns into a saddle branch. This point is called the kinetic tangent pinch [9]. The saddle branch arrives at Da = 0.0 in the binary azeotropic point between MeOH and MTBE. 

The solubility of terephthalic acid in the above-mentioned solvents is very low, which means that the acid must diffuse continuously from the solid particules to the solution where the reaction takes place. In such a case, the first question which arises is does the diffusion control the kinetics of the overall process In all cases, the authors claimed that the reaction rate is never affected by the amount of undissolved terephthalic acid and that the reaction proceeds through a chemical kinetic control. Under the experimental conditions used by Bhatia et al. the diffusion rate of terephthalic acid from the solid particles to the solution is 9.5x 10 mol cm" s at 100 °C and that of ethylene oxide from the gas phase to the liquid is 19.4 x 10" mol cm" s" . These values are far above the rate of formation of the diester(bishydroxy-ethylterephthalate), as this is only 5.84 x 10" mol cm" s" . Moreover, the independence of the reaction rate on the mass transfer effects was shown by varying the values of some parameters (e.g., ethylene oxide flow-rate, stirrer-speed, particule size, terephthalic acid charge) in a large range. 

Numerical integration (sometimes referred to as solving or simulation) of differential equations, ordinary or partial, involves using a computer to obtain an approximate and discrete (in time and/or space) solution. In chemical kinetics, these differential equations are typically the rate laws that describe the time evolution of the system. One obtains results for the mean concentrations, without any information about the (typically very small) fluctuations that are inevitably present. Continuation and sensitivity analysis techniques enable one to extrapolate from a numerically obtained solution at one set of parameters (e.g., rate constants or initial concentrations) to the behavior of the system at other parameter values, without having to carry out a full numerical integration each time the parameters are changed. Other approaches, sometimes referred to collectively as stochastic methods (Gardiner, 1990), can provide data about fluctuations, but these require considerably more computational labor and are often impractical for models that include more than a few variables. 

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