Semi-kinetic model

Computer modeling of hydrocarbon pyrolysis is discussed with respect to industrial applications. Pyrolysis models are classified into four groups mechanistic, stoichiometric, semi-kinetic, and empirical. Selection of modeling schemes to meet minimum development cost must be consistent with constraints imposed by factors such as data quality, kinetic knowledge, and time limitations. Stoichiometric and semi-kinetic modelings are further illustrated by two examples, one for light hydrocarbon feedstocks and the other for naphthas. The applicability of these modeling schemes to olefins production is evidenced by successful prediction of commercial plant data. 

The first class is a generalization of the advection-diffusion problem discussed in Section 8.3, and much of the material developed there can be reformulated to develop a realizable scheme for the GPBE. The second class is a generalization of the kinetic equation considered in Section 8.4, and has been referred to in the literature as the semi-kinetic model (Laurent et al, 2004 Laurent Massot, 2001). In the following, we will treat each class separately, although the reader will undoubtedly note many similarities between the numerical schemes. A third class of GPBE, lying between the two listed above, can also be identified wherein the scalar-dependent convection velocity has a parametric form such as... 

Ethylene oxide secondary oxidation with C-tagged ethylene oxide, to clarify the source of CO2, was done at Union Carbide but not published. This was about 10 years before the publication of Happel (1977). With very limited radioactive supply only a semi-quantitative result could be gained but it helped the kinetic modeling work. It became clear that most CO2 comes from ethylene directly and only about 20% from the secondary oxidation of ethylene oxide. 

Kinetic Model Discrimination. To discriminate between the kinetic models, semibatch reactors were set up for the measurement of reaction rates. The semi-batch terminology is used because hydrogen is fed to a batch reactor to maintain a constant hydrogen pressme. This kind of semi-batch reactor can be treated as a bateh reactor with a constant hydrogen pressme. The governing equations for a bateh reactor, using the product formation rate for three possible scenarios, were derived, as described in reference (12) with the following results ... 

Mechanisms of dissolution kinetics of crystals have been intensively studied in the pharmaceutical domain, because the rate of dissolution affects the bioavailability of drug crystals. Many efforts have been made to describe the crystal dissolution behavior. A variety of empirical or semi-empirical models have been used to describe drug dissolution or release from formulations [1-6]. Noyes and Whitney published the first quantitative study of the dissolution process in 1897 [7]. They found that the dissolution process is diffusion controlled and involves no chemical reaction. The Noyes-Whitney equation simply states that the dissolution rate is directly proportional to the difference between the solubility and the solution concentration ... 

Equilibrium studies under anaerobic conditions confirmed that [Cu(HA)]+ is the major species in the Cu(II)-ascorbic acid system. However, the existence of minor polymeric, presumably dimeric, species could also be proven. This lends support to the above kinetic model. Provided that the catalytically active complex is the dimer produced in reaction (26), the chain reaction is initiated by the formation and subsequent decomposition of [Cu2(HA)2(02)]2+ into [CuA(02H)] and A -. The chain carrier is the semi-quinone radical which is consumed and regenerated in the propagation steps, Eqs. (29) and (30). The chain is terminated in Eq. (31). Applying the steady-state approximation to the concentrations of the radicals, yields a rate law which is fully consistent with the experimental observations ... 

Kinetic modelling was then performed, based on the obtained concentration-time profiles assuming semi-empirical rate equations (Eqs. 10,11) ... 

Glaze, W Kang, J. Advanced oxidation processes. Test of a kinetic model for the oxidation of organic compounds with ozone and hydrogen peroxide in a semi batch reactor, Industrial and Engineering Chemistry Research, 1989 28, 1580-1587. 

This section will begin by looking at how thermodynamic and kinetic modelling has been combined to understand time-temperature-transformation diagrams in steels. The woric, for the most part, is semi-empirical in nature, which is forced upon the topic area by difficulties associated with the diffusional transformations, particularly where nucleation aspects have to be considered. The approaches have considered how best to predict the time/temperature conditions for austenite to... 

As appears from the examination of the equations (giving the best fit to the rate data) in Table 21, no relation between the form of the kinetic equation and the type of catalyst can be found. It seems likely that the equations are really semi-empirical expressions and it is risky to draw any conclusion about the actual reaction mechanism from the kinetic model. In spite of the formalism of the reported studies, two observations should be mentioned. Maatman et al. [410] calculated from the rate coefficients for the esterification of acetic acid with 1-propanol on silica gel, the site density of the catalyst using a method reported previously [418]. They found a relatively high site density, which justifies the identification of active sites of silica gel with the surface silanol groups made by Fricke and Alpeter [411]. The same authors [411] also estimated the values of the standard enthalpy and entropy changes on adsorption of propanol from kinetic data from the relatively low values they presume that propanol is weakly adsorbed on the surface, retaining much of the character of the liquid alcohol. 

This example demonstrates how reaction calorimetry in combination with IR-ATR spectroscopy can be used to discriminate between different postulated reaction models, and to determine the kinetic and thermodynamic parameters for the selected model. In practical applications, when different (semi-) empirical models can be postulated, model discrimination is crucial. 

Eqs. 3-4 are amenable to semi-analytical solution techniques because of the linear form. The use of more complex kinetic models (e.g., intraaggregate diffusion) has not been attempted, in part because the above models have proved adequate to describe the available data sets, and in part because of a limited understanding of the geometry of the soil/bentonite matrix (gel formation and the resulting diffusion geometry). 

Nevertheless, at present a distinct tendency is observed not to restrict oneself to the semi-empirical equations, but to apply kinetic models corresponding to the physico-chemical substantiated mechanisms. As a rule, modifications of the well-known adsorption (L-H) and impact (E-R) mechanisms are used. As usual, the complex mechanism always has a constituent accounting for the adsorption mechanism. 

This simple analysis is semi empirical it is not a description of the diffusion limited reaction within the crystals but allows one to take into account both phenomena, in order to provide kinetic models for FCC reactor description [11]. Experimental results on the three feedstocks are shown in figure 1, with the deactivation function determined according to the method described in [10]. Curves are calculated from equation (3) after fitting E and F. These values are reported in table 2. 

Ross and Shu [38], discussing the computer modelling of hydrocarbon pyrolysis for olefin production, classify reaction models in four categories in order of increasing sophistication empirical, semi-kinetic, stoichiometric and mechanistic. Most concepts of this classification are included in Table 3 with, however, a more classical meaning of the word stoichiometry. 

This section proposes the use of a semi-Markov model with Erlang- and phase-type retention-time distributions as a generic model for the kinetics of systems with inhomogeneous, poorly stirred compartments. These distributions are justified heuristically on the basis of their shape characteristics. The overall objective is to find nonexponential retention-time distributions that adequately describe the flow within a compartment (or pool). These distributions are then combined into a more mechanistic (or physiologically based) model that describes the pattern of drug distribution between compartments. The new semi-Markov model provides a generalized compartmental analysis that can be applied to compartments that are not well stirred. 

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