A Two-Zone Model

The underlying idea for such a model is that only a fraction of the liquid flows in a more or less ordered way through the packing, while at each height there is a stagnant zone in which the liquid is well mixed and that exchanges mass with the flowing fraction. 

The continuity equation for A in the gas phase is either the same as in (14.2.4-1) or is simplified by neglecting the effective diffusion term. The continuity equations for A in the liquid phases, neglecting effective diffusion, may be written 

For the well-mixed liquid in the corresponding slice of stagnant liquid  

The transfer coefficient between the two zones, kj, and that between the stagnant fluid and the solid, kj contain interfacial areas that are not well established presently. 

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A genera steady-state balance calculation of a two-zone model is presented in Figs. 8.6-8.7 and Eqs. (8.10H8.17). 

Mundt presents a two-zone model for the calculation of temperature gradient within a stratification strategy. 

Sandberg et al. present a two-zone model for a zoning strategy and active displacement air distribution method. 

Figures 8.33 and 8.34 describe a two-zone model application of the zoning strategy where all the main variable parameters are presented. Figure 8.33 (temperature model describes the accumulation of heat and Fig. 8.34 (concentration model) the accumulation of contaminants. After solving for the temperatures, heat flow s, and airflows, contaminant concentrations can lie calculated. The models are here determined for stationary loads, airflow rates, and indoor/outdoor conditions, but they can be developed also for dynamic simulations.

The indirect mechanism postulates a two-zone model with strongly exothermic CH4 combustion to H20 and C02 (Eq. 3.6) at the catalyst entrance, followed by strongly endothermic steam (see Eq. 3.3) and C02 reforming downstream (Eq. 3.7) ... 

Wu and Lee 42 indicated that the free chloride ions on the active site (measured by Volhard analysis) were at only 50-70 i, of the amount of immobilized content (measured by element analysis). The results of the Volhard analysis method determined the free chloride ions in the bulk solution by the AgNOt titration method. Their results implicHl that the active site in the resin could not react completely w ith organic reactant in the duration of triphase reaction. According to the experimental results, this reaction is a two-zone model (or shell-core model), with the reaction (K-cuiring in the shell zone, and not in the core zone. Therefore, the triphasic reactitm mechanism and the swollen type of the resin can be given in Figure 4. This mechanism can help to understand the reaction phenomenon in the triphasc rcaciion. 

Combined quantum mechanical/molecular mechanical methods are not, of course, restricted to studies of reactions but can also be used to study association processes and conformational transitions. Most implementations use a two-zone model as described above, but Morokuma and colleagues have described a multilayered approach called ONIOM [Svensson et al. 1996]. ONIOM is a particularly apt name given that a typical calculation is constructed from a series of layers For example, a three-layer ONIOM calculation on the Diels-Alder reaction involved an inner core treated with the B3LYP density functional approach, the intermediate layer with a Hartree-Fock level of theory and the outer layer with MM3. A particular feature of ONIOM and its related methods is that they provide rigorous gradients and second derivatives, so enabling properties such as vibrational frequencies to be calculated [Dapprich et al. 1999]... 

In Figure (9), temperature and species profile plots obtained in an investigation of autoignition in the end-gas of an Sl-engine, known as knock, as presented by Soyhan et al. ( yhan et al., 2000), are shown. The calculations are obtained in employing a two-zone model (burnt and unburnt zones), the detailed mechanism for iso-octane and n-heptane mixtures... 

In some applications, a simple CSTR is not sufficient to describe the residence time distribution of particles. Habermann et al. (1998) described a metallurgical process where the fluidized bed showed a stagnant zone and a two-zone model with a CSTR and a stagnant dead-zone fitted the data much better. For the resi-... 

For each region a mean value of the void fraction was calculated and a hydraulic radius was defined which was used in a pressure drop correlation. Martin [20] divided the bed into two regions a wall and a bulk region. He calculated for both different flow rates and a different rate of heat transfer. Carbonell [2] also used a two zone model for his analysis of the dispersion phenomena. In more recent work Vortmeyer et al. [5>6] tried to use the complete radial void fraction profile, and so did Chang [3]. They followed the same itinerary outlined by Lerou and Froment [l] and Marivoet et al. [2l]. Starting from the void fraction profile the radial velocity profile is calculated. With both profiles the effective thermal conductivity is established and the temperature and concentration profiles can be calculated by means of a two dimensional pseudo homogeneous model for the reactor. 

Such a two-zone model, incorporating variable times of contact... 

Sales et al. (2005) modeled the gas phase dynamic behavior of a three-phase fluidized bed reactor to evaluate the behavior of a gas tracer (methane) in the reactor. A two-zone model was proposed for the gas phase based on the operational behavior of the system. The dynamic model assumed that the total gaseous volume inside the reactor was distributed in two zones, represented by a tubular reactor of volume connected in series with a CSTR of volume The volumes of the two zones are related as... 

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