Heterogeneous Dispersion Models

The corresponding heat or temperature equation can be derived following a similar procedure as adopted formulating the single phase equation (1.299). The result is  

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LPCVD Reactor Models. First-Order Surface Reaction. The traditional horizontal-wafer-in-tube LPCVD reactor resembles a fixed-bed reactor, and recent models are very similar to heterogeneous-dispersion models for fixed-bed reactors (21,167,213). To illustrate CVD reactor modeling, this correspondence can be exploited by first considering a simple first-order surface reaction in the LPCVD reactor and then discussing complications such as complex reaction schemes, multicomponent diffusion effects, and entrance phenomena. 

The collocation methods can be shown to give rise to symmetric, positive definite coefficient matrices that is characterized with a acceptable condition number for diffusion dominated problems or other higher order even derivative terms. For convection dominated problems the collocation method produces non-symmetric coefficient matrices that are not positive definite and characterized with a large condition number. This method is thus frequently employed in reactor engineering solving problems containing second order derivatives of smooth functions. A t3q)ical example is the pellet equations in heterogeneous dispersion models. 

Figure 6.7 A dispersion model that incorporates spatial heterogeneity for the gastrointestinal absorption processes, qo denotes the administered dose and

Recently, a novel convection-dispersion model for the study of drug absorption in the gastrointestinal tract, incorporating spatial heterogeneity, was presented [182]. The intestinal lumen is modeled as a tube (Figure 6.7), where the concentration of the drug is described by a system of convection-dispersion partial differential equations. The model considers ... 

However, these two models assume either perfect mixing conditions (well-stirred model) or no mixing at all (parallel tube model) and cannot explain several experimental observations. Therefore, other approaches such as the distributed model [268], the dispersion model [269], and the interconnected tubes model [270,271] attempt to capture the heterogeneities in flow and an intermediate level of mixing or dispersion. Despite numerous comparisons [264,265,272-... 

Wilson and Liu showed that both location and travel time probabilities can be calculated directly, using a backward-in-time version of traditional continuum advection-dispersion modeling. In addition, they claimed that by choosing the boundary conditions properly, the method can be readily generalized to include linear adsorption with kinetic effects and 1st order decay. An extension of their study for a 2D heterogeneous aquifer was reported in Liu and Wilson [39]. The results for travel time probability are in very close agreement with the simulation results from traditional forward-in-time methods. 

The pseudo-homogeneous fixed bed dispersion models are divided into three categories The axial dispersion model, the conventional two-dimensional dispersion model, and the full two-dimensional axi-symmetrical model formulation. The heterogeneous fixed bed dispersion models can be grouped in a similar way, but one dimensional formulations are employed in most cases. 

The enrichment of the slow dissolving component, B, in an alloy surface under simultaneous dissolution conditions may be rationalized by a model of alloy dissolution that is based on the simplifying assumptions (1) that a homogeneous solid solution may be described as a heterogeneous dispersion of atomic dimensions with area fraction (surface mole fraction) X j for component j, and (2) that the alloy components dissolve independently. The partial current density ij of an alloy component j will then be given by ij = i -X j, where i is the current density of the pure metal, j, and for a binary alloy A-B, the total current density of alloy dissolution. 

The developed dynamic reactor model for the simulation studies of the unsteady-state-operated trickle-flow reactor is based on an extended axial dispersion model to predict the overall reactor performance incorporating partial wetting. This heterogeneous model consists of unsteady-state mass and enthalpy balances of the reaction components within the gas, liquid and catalyst phase. The individual mass-transfer steps at a partially wetted catalyst particle are shown in Fig. 4.5. 

Under these circumstances, a general stationary heterogeneous dispersion (PD-)model for an irreversible catalytic second order reaction between a gaseous and a liquid reactant in dimensionless form consists of the balance equations shown in Fig. 18. In this model the whole fluiddynamics are lumped into a single parameter, i.e. the Bodenstein number, here based on the reactor length. 

Fig, 18 A general steady state isothermal heterogeneous dispersion (PD-)model for multiphase catalytic reactors... 

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