Two-dimensional, one-phase model

In the steady-state operation of the OXITOX reactor, pelletized solid of catalytieally aetivated sodium earbonate slides down a Silo type reaetor. Counter-current to the solid flow, the polluted air rises through the sliding bed of solids. At reaction temperature the following reaetion oecurs  

The trichloroethylene is oxidized, the gaseous products are removed by the flowing air, and the ehlorine is eaptured by the solid soda and forms salt. The solid salt is removed by diseharging the used OXITOX at the bottom of the reaetor. This is a relatively slow reaetion and the central interest is in removing the last traees of toxic chlorinated compounds (for which TCE is only a model eompound), therefore a very simple model was used. Based on conservation prineiples, it was assumed that chloride removed from the gas phase ends up in the solid phase. This was proven in several material balanee ealeulations. No HCl or other ehlorinated compound was found in the gas phase. The eonsumption rate for TCE was expressed as  

Taking the upward direction as positive, when both Fy 0 and Fs 0, represents a co-current downflow operation, if Fs 0, but Fy 0 means a counter-current operation. The rest of the equation defining the system can be seen in Berty (1997). Integration was done by the Romberg method as is used in Mathcad PLUS 6 (1996) software. 

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Because of the numerical problems mentioned above a spatially one-dimensional two-phase model has been used to simulate the more rapid transients. The model equations for Figure 8 and the parameters used are given in Table I. In Figures 5,6 the decomposition reaction to methane has also been considered. 

Theoretical and experimental results on deactivation have been summarized in two reviews by Butt (.1,2). Previous work of particular interest to the present study has been done by Blaum (3) who used a one-dimensional two-phase model to explore the dynamic behavior of a deactivating catalyst bed. Butt and cowotkers (b,5,6)have performed deactivation studies in a short tubular reactor for benzene hydrogenation for both adiabatic and nonadiabatic arrangements. They experimentally observed both the standing (6) and travelling (4) deactivation wave. Hlavacek... 

Kiel et al. [36] used one-dimensional two-phase model to calculate gas-solids mass transfer coefficients. Basic assumptions for the model were (i) gas-solid mass transfer is the only resistance for adsorption and (ii) the effective area for mass transfer is equal to the external siuface area of the spheres. Radial effects were neglected due to the good radial distribution properties of the regularly stacked packing. Axial dispersion in the gas phase was also estimated from the experimental results presented by Roes and van Swaaij [29,30]. 

A typical one-dimensional two-phase model for a membrane-assisted FB reactor can be used for the simulation of the FBMR for hydrogen production via methane reforming. A schematic representation of the gas flows between the compartments of the bubble and emulsion phases is depicted in Figure 3.12. The model main assumptions are ... 

As reactor model a conventional one-dimensional two-phase model with axial dispersion of heat and mass is used [4]. It consists of energy balances for the gas and the solid phase and a mass balance for the organic pollutant. The equations are given as follows ... 

In reality, the slip velocity may not be neglected (except perhaps in a microgravity environment). A drift flux model has therefore been introduced (Zuber and Findlay, 1965) which is an improvement of the homogeneous model. In the drift flux model for one-dimensional two-phase flow, equations of continuity, momentum, and energy are written for the mixture (in three equations). In addition, another continuity equation for one phase is also written, usually for the gas phase. To allow a slip velocity to take place between the two phases, a drift velocity, uGJ, or a diffusion velocity, uGM (gas velocity relative to the velocity of center of mass), is defined as... 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

Wulff, W, 1978, Lump-Parameter Modeling of One-Dimensional Two-Phase Flow, in Transient Two-Phase Flow, Proc. 2nd Specialists Meeting, vol. 1, pp. 191-219, OECD Committee on Safety of Nuclear Installations, Paris. (3)... 

Divisek et al. presented a similar two-phase, two-dimensional model of DMFC. Two-phase flow and capillary effects in backing layers were considered using a quantitatively different but qualitatively similar function of capillary pressure vs liquid saturation. In practice, this capillary pressure function must be experimentally obtained for realistic DMFC backing materials in a methanol solution. Note that methanol in the anode solution significantly alters the interfacial tension characteristics. In addition, Divisek et al. developed detailed, multistep reaction models for both ORR and methanol oxidation as well as used the Stefan—Maxwell formulation for gas diffusion. Murgia et al. described a one-dimensional, two-phase, multicomponent steady-state model based on phenomenological transport equations for the catalyst layer, diffusion layer, and polymer membrane for a liquid-feed DMFC. 

The most reliable methods for fully developed gas/liquid flows use mechanistic models to predict flow pattern, and use different pressure drop and void fraction estimation procedures for each flow pattern. Such methods are too lengthy to include here, and are well suited to incorporation into computer programs commercial codes for gas/liquid pipeline flows are available. Some key references for mechanistic methods for flow pattern transitions and flow regime-specific pressure drop and void fraction methods include Taitel and Dukler (AIChEJ., 22,47-55 [1976]), Barnea, et al. (Int. J. Multiphase Flow, 6, 217-225 [1980]), Barnea (Int. J. Multiphase Flow, 12, 733-744 [1986]), Taitel, Barnea, and Dukler (AIChE J., 26, 345-354 [1980]), Wallis (One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969), and Dukler and Hubbard (Ind. Eng. Chem. Fun-dam., 14, 337-347 [1975]). For preliminary or approximate calculations, flow pattern maps and flow regime-independent empirical correlations, are simpler and faster to use. Such methods for horizontal and vertical flows are provided in the following. 

Morookact al. (M42) further solved the impulse response of a tracer gas for a fluidized catalyst bed according to the one-dimensional two-phase diffusion model (VI), where the influence of the particle capacitance effect was considered under the assumption of local adsorption equilibrium. The equations of continuity for the tracer gas are ... 

The influence of longitudinal dispersion on the extent of a first-order catalytic reaction has been studied by Kobayashi and Arai (K14), Furusaki (F13), van Swaay and Zuiderweg (V8), and others. They use the one-dimensional two-phase diffusion model, and show that longitudinal dispersion of the emulsion has little effect when the reaction rate is low. Based on the circulation flow model (Fig. 2) Miyauchi and Morooka (M29) have shown that the mechanism of longitudinal dispersion in a fluidized catalyst bed is a kind of Taylor dispersion (G6, T9). The influence of the emulsion-phase recirculation on the extent of reaction disappears when the term tp defined by Eq. (7-18) (see Section VII) is greater than about 10. For large-diameter beds, where p does not satisfy this restriction, their general treatment includes the contribution of Taylor dispersion for both the reactant gas and the emulsion (M29). 

Kiel et al. [41] proposed a model, which predicts the performance of a full-scale gas-flowing solids-fixed bed absorber, for flue gas desulfurization. An one-dimensional, two-phase axially dispersed model has been applied. The model was derived from separate mass and heat balances for both gas and (porous) solid phases in the case of noncatalytic gas-sohd reaction, which is first order in the gas phase. 

A computer model was developed to estimate the oil and water relative permeability exponential functions from displacement experiments. The model consist of an optimization algorithm, a one-dimensional two phase flow simulator with capillary effects, and a simple Buckley Leverett non-capillary flow simulator. Using experimental data and the model, the effect of rate on the parameters of relative permeability functions were studied. The following conclusions can be delivered from the results of this study. 

The preponderant majority of contemporary mathematical models which are used to analyze a wide variety of steady, adiabatic, one-dimensional two-phase flows, with the single space coordinate z, can be reduced to the standard form... 

Vortmeyer D, Schaefer RJ. Equivalence of one- and two-phase models for heat transfer processes in packed beds one dimensional theory. Chemical Engineering Science 1974 29 485-491. 

In addition, the numerical models can be used in order to understand the overall effect of CO poisoning on other transport phenomena, such as liquid water transport. In a study by Wang and Chu [24], they developed a transient, one-dimensional, two-phase numerical model of the electrolyte membrane and anode and cathode catalyst layers. Their model was used to look into the effect of CO poisoning on the water distribution in the catalyst layers and the electrolyte membrane. With 100% H2 (i.e., the hydrogen feed was not dilute), 10 ppm CO level, and a cell voltage of 0.6 V, they investigated the liquid water saturation in the catalyst layers and the water content in... 

In the smdy by Solsvik and Jakobsen [140], a one-dimensional two-fluid model describing gas-solid flows with chemical reactions in bubbling bed reactors was derived. The reactor system was modeled in terms of mass, species mass, heat and momentum balances for each of the phases in the Eulerian reference frame. The governing equations describing the reactive flow are presented in the sequent. 

In order to enlarge the scope of the code applicability to reactor cases, the implementation of a one-dimensional two-phase flow thermalhydraulics with 2 non-condensible gases is underway. The set of 8 balance equations (4 mass, 2 momentum and 2 ener equations) is completed by the physical grids of the French thermalhydraulics CATHARE 2 code. The introduction of this 2 phase flow model is carried out in such a way that the coupling between ICARE2 and CATHARE 2 codes will be possible. 

The sections that now follow report a two-dimensional, two-phase formulation of the particle bed model (Chen et al., 1999), and apply it to a number of key problems in fluidization dynamics. The generalization follows naturally from the one-dimensional, two-phase treatment described in Chapter 11. It remains fully predictive, no arbitrary or adjustable parameters being introduced. 

Ishii (1977) One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase regimes. AML Report ANL-77-47 Ide H, Matsumura H, Tanaka Y, Fukano T (1997) Flow patterns and frictional pressure drop in gas-liquid two-phase flow in vertical capUlary channels with rectangular cross section, Trans JSME Ser B 63 452-160... 

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