Quasi-single-phase model

The calculation principle on which the assessment of design for such reactors is based is a substitution of the multi-phase reaction system by a quasi-single-phase model. In two-phase systems both reactants have to get into contact at a certain place. Consequently a reaction and a transport phase are distinguished. If the mass transfer rate from the transport to the reaction phase is veiy fast compared to the actual reaction rate, the process in total is dominated by the reaction kinetics. In order to discriminate this situation from one taking the mass transfer into account, it is referred to as micro-kinetically dominated In this ease all formal kinetic laws presented for homogeneous systems may be applied directly. 

In the quasi-single-fluid models, the rising gas-liquid mixture is treated like a homogenous fluid of reduced density, and one set of continuity and momentum equations is solved for the two-phase mixture. The quasi-single-phase modeling technique has been relatively more popular and has been used extensively by Szekely and co-workers [34,42,43] and Guthrie and co-workers [1,2,15,25,44 7]. The k—s model is often used to represent turbulence. In most applications, the void fraction distribution is assumed a priori rather than being solved for, and this limits the predictive capability of these models. 

Although the / — turbulence model has been applied extensively [1,2,44, 61] for modeling fluid flow in gas-stirred ladle systems, some fluid model studies [50] indicate that the / — model cannot accurately simulate the distribution of various turbulence parameters in the gas-stirred system. Despite this, it is demonstrated that the — model has been reasonably successful in predicting the bulk liquid flows as these are largely dominated by inertial rather than turbulence viscous forces. It is important to note here that the inadequacy of the A — model to simulate turbulence in the gas-stirred reactors has been attributed to the quasi-single-phase modeling technique [40], since exact two-phase computational procedures have been shown to produce fairly accurate estimates of turbulence parameters in the system. 

Several attempts to describe replication-mutation networks by stochastic techniques were made in the past. We cannot discuss them in detail here, but we shall brieffy review some general ideas that are relevant for the quasispecies model. The approach that is related closest to our model has been mentioned already [51] the evolutionary process is viewed as a sequence of stepwise increases in the populations mean fitness. Fairly long, quasi-stationary phases are interrupted by short periods of active selection during which the mean fitness increases. The approach towards optimal adaptation to the environment is resolved in a manner that is hierarchical in time. Evolution taking place on the slow time scale represents optimization in the whole of the sequence space. It is broken up into short periods of time within which the quasi-species model applies only locally. During a single evolutionary step only a small part of sequence space is explored by the population. There, the actual distributions of sequences resemble local quasispecies confined to well-defined regions. Error thresholds can be defined locally as well. 

Recall that the models used in the developments just presented are based on the quasi-continuum approach. As already noted, this means that the gas-solid quasicontinuum is regarded as a single phase with properties of its own. These are not thermodynamic properties but effective properties that can be used to analyze transport phenomena within the continuum, as in any homogeneous system. 

Castillejos et al. [52,53] solved the quasi-single-model equations considering that the ladle is occupied by a single-phase fluid with spatially variable density. Gas void fraction is prescribed with an empirical model developed from the experimental data of Castillejos and Brimacombe [18,52,53] thus ... 

Statistical methods represent a background for, e.g., the theory of the activated complex (239), the RRKM theory of unimolecular decay (240), the quasi-equilibrium theory of mass spectra (241), and the phase space theory of reaction kinetics (242). These theories yield results in terms of the total reaction cross-sections or detailed macroscopic rate constants. The RRKM and the phase space theory can be obtained as special cases of the single adiabatic channel model (SACM) developed by Quack and Troe (243). The SACM of unimolecular decay provides information on the distribution of the relative kinetic energy of the products released as well as on their angular distributions. 

This study firstly aims at understanding adsorption properties of two HSZ towards three VOC (methyl ethyl ketone, toluene, and 1,4-dioxane), through single and binary adsorption equilibrium experiments. Secondly, the Ideal Adsorbed Solution Theory (IAST) established by Myers and Prausnitz [10], is applied to predict adsorption behaviour of binary systems on quasi homogeneous adsorbents, regarding the pure component isotherms fitting models [S]. Finally, extension of adsorbed phase to real behaviour is investigated [4]. 

As for the previous example described in Sect. 3.2.1, the relaxation of the magnetization has been studied using combined ac (Fig. 6a) and dc (Fig. 6b) measurements. In order to extract the relaxation time of the system (r), the obtained frequency dependence of the in-phase x and out-of-phase x" susceptibilities and furthermore the Cole-Cole plots (x" vs. x plot) were fitted simultaneously to a generalized Debye model (solid lines in Figs. 6a and 6b). The fact that the found a parameters of this model are less than 0.06, indicates that the system is close to a pure Debye model with hence a single relaxation time. This indication is confirmed by the quasi-exponential decay of the magnetization observed between 1.8 and 0.8 K (Fig. 6b). 

In this paper we have tried to describe the most important steps on the way from the simple tight-binding approximation to the simple fluctuation model for quasi-one-dimensional conductors. A major crossing on this way is certainly traversed in earlier sub-sections, where are given the arguments in favour of the single-order parameter phase transition theory for high-temperature quasi-one-dimensional conductors. 

Because the solute flux is small relative to total blood flow, we can model a single hollow fiber as a straight pipe with walls permeable to certain solutes, so at quasi-steady state in the flowing blood phase, the... 

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