Explicit model diffusion kinetics

Computed results from this model are compared to actual kiln performance in Table VI and the operating conditions taken from kiln samples are given in Table VII. There are no unit factors or adjustable parameters in this model. As with the explicit model, all kinetic data are determined from laboratory experiments. Values of the frequency factors and activation energies are given in Table VIII. Diffusivity values are also included. The amount of fast coke was determined from Eq. (49). With the exception of the T-B (5/12) survey, the agreement between observed and computed values of CO, CO2, and O2 is very good considering that there are no adjustable parameters used to fit the model to each kiln. In the kiln survey T-212/10, the CO conversion activity of the catalyst has been considerably deactivated and a different frequency factor was used in this simulation. 

Solutions for this type of kinetics can only be achieved numerically. Model calculations with constant kinetic parameters have been made [H. Wiedersich, et al. (1979)], however, the modeling of realistic transport (diffusion) coefficients which enter into the flux equations is most difficult since the jump rate vA vB. Also, the individual point defects have limited lifetimes which determine the magnitude of correlation factors (see Section 5.2.2). Explicit modeling for dilute or non-dilute alloys can be found in [A.R. Allnatt, A.B. Lidiard (1993)]. 

The mathematical formulation of the PBPK model is dependent on several factors routes of intake of a chemical or sites of drug administration, target tissues of interest, physiological components to be explicitly modeled (kinetically important tissues and organs and the linkages between them), transport processes of the chemical (flow, diffusion, disposition, clearance, etc.), and metabolic processes involved. 

As shown in Example 22-3, for solid particles of the same size in BMF, the form of the reactor model resulting from equation 22.2-13 depends on the kinetics model used for a single particle. For the SCM, this, in turn, depends on particle shape and the relative magnitudes of gas-film mass transfer resistance, ash-layer diffusion resistance and surface reaction rate. In some cases, as illustrated for cylindrical particles in Example 22-3(a) and (b), the reactor model can be expressed in explicit analytical form additional results are given for spherical particles by Levenspiel(1972, pp. 384-5). In other f l cases, it is convenient or even necessary, as in Example 22-3(c), to use a numerical pro-... 

The variation of efficiencies is due to interaction phenomena caused by the simultaneous diffusional transport of several components. From a fundamental point of view one should therefore take these interaction phenomena explicitly into account in the description of the elementary processes (i.e. mass and heat transfer with chemical reaction). In literature this approach has been used within the non-equilibrium stage model (Sivasubramanian and Boston, 1990). Sawistowski (1983) and Sawistowski and Pilavakis (1979) have developed a model describing reactive distillation in a packed column. Their model incorporates a simple representation of the prevailing mass and heat transfer processes supplemented with a rate equation for chemical reaction, allowing chemical enhancement of mass transfer. They assumed elementary reaction kinetics, equal binary diffusion coefficients and equal molar latent heat of evaporation for each component. 

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

Several different models were proposed for the slow NOx storage process, while only few details and approximate models are available for the highly transient NOx reduction within the rich phase, lasting only several seconds. The models can be divided into two groups, depending on whether the internal diffusion in the particles of the NOx storage material is considered explicitly, or this effect is included implicitly into the evaluated kinetic parameters. The models can be further differentiated by the level of complexity for the reaction kinetics description, i.e., either (simplified) microkinetic scheme or the global kinetics. 

In rate-based multistage separation models, separate balance equations are written for each distinct phase, and mass and heat transfer resistances are considered according to the two-film theory with explicit calculation of interfacial fluxes and film discretization for non-homogeneous film layer. The film model equations are combined with relevant diffusion and reaction kinetics and account for the specific features of electrolyte solution chemistry, electrolyte thermodynamics, and electroneutrality in the liquid phase. 

Often, the same expressions from this model are applied to the reductions of pellets, in which cases such structural factors as particle size distribution, porosity and pore shape, its size distribution, etc. should really affect the whole kinetics. Thus, the application of this model to such systems has been criticised as an oversimplification and a more realistic model has been proposed [6—12,136] in which the structure of pellets is explicitly considered to consist of pores and grains and the boundary is admitted to be diffusive due to some partly reduced grains, as shown in Fig. 4. Inevitably, the mathematics becomes very complicated and the matching with experimental results is not straightforward. To cope with this difficulty, Sohn and Szekely [11] employed dimensional analysis and introduced a dimensionless number, a, given by... 

It is further noted that the use of interfacial mass flux weighted transfer terms is generally not convenient treating multicomponent reactive systems, because the phase change processes are normally not modeled explicitly but deduced from the species composition dependent joint diffusive and convective interfacial transfer models. Moreover, the rigorous reaction kinetics and thermodynamic models of mixtures are always formulated on a molar basis. 

Analysis of receptor/ligand interactions is made much more complicated by explicit consideration of these physical features. Without such considerations, we are often able to model receptor phenomena with first-order ordinary differential equations based on straightforward mass-action kinetic species balances. When diffusion and probabilistic effects are taken into account, the models can easily give rise to partial differential equations, second-order ordinary differential equations, extremely large sets of first-order ordinary differential equations, and/or probabilistic differential equations. 

This observation constitutes the basic idea of the local equilibrium model of Prigogine, Nicolis, and Misguich (hereafter referred to as PNM). One considers the case of a spatially nonuniform system and deduces from (3) an integral equation for the pair correlation function that is linear in the gradients. This equation is then approximated in a simple way that enables one to derive explicit expressions for all thermal transport coefficients (viscosities, thermal conductivity), both in simple liquids and in binary mixtures, excluding of course the diffusion coefficient. The latter is a purely kinetic quantity, which cannot be obtained from a local equilibrium hypothesis. 

Another, less widely appreciated idealization in chemical kinetics is that phenomena take place instantaneously—that a change in [A] at time t generates a change in [fi] time t and not at some later time t + z. On a microscopic level, it is clear that this state of affairs cannot hold. At the very least, a molecular event taking place at point x and time t can affect a molecule at point x only after a time of the order of x — x f jlD, where D is the relevant diffusion constant. The consequences of this observation at the macroscopic level are not obvious, but, as we shall see in the examples below, it may sometimes be useful to introduce delays explicitly in modeling complex reaction networks, particularly if the mechanism is not known in detail. 

Model formulation usually implies mass balances where the interplay between chemical reactions and transport phenomena (i.e., monomer and oligomer diffusion) is considered. The literature offers a wide variety of models with different degrees of complexity. An autocatalytic effect is not always explicitly included or a diffusivity increase because of hydrolysis onset. Degradation is usually described with pseudo-first order kinetics (where it is assumed that water concentration is constant) or second order kinetics, where the hydrolysis rate depends on the ester bond and water concentration, or on the ester bond and acid concentration when autocatalysis in considered. Also 1.5th and 2.5th order kinetics have been proposed (Siparsky et al., 1998 Lyu et al., 2007). 

An improved model in regard to the simple model presented in Section 9.4.2 has been cut out for more realistic conditions [42]. Not only is the swelling process explicitly tied to an ionic pressure, but there are also other improvements the transport description includes the solvent flow as weU as the dynamics of charged species, the dependence of diffusion coefficients on the gel density is accounted for, and, finally, the kinetic toy model is replaced by a reaHstic model of a spatially bistable reaction. We consider that the functional unit HA of the polyacid has a unique weak-acid function. 

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