Two dimensional homogeneous model

Also, heat transfer in three-phase fixed-bed reactors has been investigated using a two-dimensional homogeneous model with two parameters [94, 96]—the bed radial effective thermal conductivity and the heat transfer coefficient at the wall ... 

Recorded temperature profiles from the monotube pilot plant in Figure 3.6 are shown in Figure 3.8 [393]. The simulation uses the two-dimensional homogeneous model using the outer tube-wall temperature... 

Two-dimensional homogeneous model with porosity and void fraction profile (2D-HOM-PVP). Radial conversion (a) and temperature (b) profiles at a given distance in the bed and for Rsp =175.2 = 2 1 = 0.07 [Papageorgiou and Froment, 1995]. 

The effects due to the finite size of crystallites (in both lateral directions) and the resulting effects due to boundary fields have been studied by Patrykiejew [57], with help of Monte Carlo simulation. A solid surface has been modeled as a collection of finite, two-dimensional, homogeneous regions and each region has been assumed to be a square lattice of the size Lx L (measured in lattice constants). Patches of different size contribute to the total surface with different weights described by a certain size distribution function C L). Following the basic assumption of the patchwise model of surface heterogeneity [6], the patches have been assumed to be independent one of another. 

This equation may be used as an appropriate form of the law of energy conservation in various pseudo homogeneous models of fixed bed reactors. Radial transport by effective thermal conduction is an essential element of two-dimensional reactor models but, for one-dimensional models, the last term must be replaced by one involving heat losses to the walls. 

The appropriate mathematical model for solute transport originating from the dissolution of a perchloroethylene (PCE) pool located onto a bedrock within a two-dimensional, homogeneous, water saturated aquifer in the presence of dissolved humic substances, as illustrated in Fig. 8, consists of three coupled transport equations. One equation describing the transport of the solute originating from the dissolving PCE pool in the presence of dissolved humic substances, another equation describing the transport of dissolved humic substances, and the third equation describing the transport of solute-humic particles. 

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. 

This pseudo-homogeneous two-dimensional dispersion model formulation is strictly not consistant with the cross sectional averaging procedure outlined in sect 3.4.6 and sect 1.2.7 and should be treated with prudence and/or avoided. 

The pseudo-homogeneous two-dimensional dispersion model, consisting of (11.11) to (11.18) and the pressure drop relations (11.3) and (11.4), was solved for the methanol production process under non-adiabatic conditions. 

Fig. 11.4. Species mole fractions and temperature profiles predicted by the pseudo-homogeneous two-dimensional dispersion model. Note, as a 2D axisymmetric tube has been simulated, the model is solved for positive r-values only. Reprinted with permission by Elsevier [5].

A pseudo-homogeneous, two-dimensional reactor model for membrane reactors consists of the total gas-phase continuity and Navier-Stokes equations augmented with gas-phase component mass balances and the overall energy balance. 

The two-dimensional Ising model permits the evaluation of the nucleation rate, beyond an unpredictable proportionality factor, expressed in the unit of numbers of critical nuclei formed per Monte Carlo step and lattice site. Figure 10.12 compares both theoretical and experimental results for crystallization of lysozyme. The values of span over two orders of magnitude from 9.1 x 10 /cycle site at low roughness (r = 1.2) to 5.2 X 10" /cycle site at r = 1.6. As a reference. Sear obtained a nucleation rate on an impurity of 6 spins on the order of 10 /cycle site, compared to a homogeneous nucleation rate of 10 /cycle site (Sear 2006). 

Other applications of convective reforming are associated with recovery of the process gas heat in ammonia and methanol plants [395] [479] and lately also in ATR-based syngas units for GTL plants as discussed in Chapter 2. Design and simulation of such a reformer is shown in [519], where the two-dimensional heterogeneous model has been applied using the kinetics in [525]. A homogeneous model has also been used, and almost identical temperature profiles have been foimd. 

Under these conditions it was expected that pore diffusion would be a limiting step and therefore effective rate parameters (see also Fig. 17) had to be used in the simulation. The simulation was based on a pseudo-homogeneous two-dimensional reactor model using the standard routine program FIBSAS /I9/. 

If we throw into the pot the further assumption of plug-flow, which seems a reasonable ideal in long and narrow tubes, and stir, what matures is a two-dimensional homogeneous continuum model, containing two parameters which we would prefer to determine by experiment. 

Fig. 7.3 Two-dimensional calculation models of the homogeneous cores (1/4 core) (in centimeters)...

In Chap. 7, the investigation on combustion stabihty is extended to propane-fueled catalytic microreactors, using the catalytic and gas-phase chemical reaction schemes of propane combustion on platinum proposed and validated in Chap. 4. The steady hetero-Zhomogeneous combustion of lean propaneZair and methaneZair mixtures in a platinum-coated, catalytic plane channel-flow microreactor were investigated at pressures of 1 and 5 bar, channel heights of 1.0 and 0.3 mm, and wall thermal conductivities of 2 and 16 WZmK. Stability limits were assessed as a function of fuel type, inlet velocity, and imposed external heat losses. Parametric studies were performed with a full-eUiptic, two-dimensional numerical model employing detailed gas-phase (homogeneous) reaction schemes for both fuels. 

To present briefly the different possible scenarios for the growth of multilayer films on a homogeneous surface, it is very convenient to use a simple lattice gas model language [168]. Assuming that the surface is a two-dimensional square lattice of sites and that also the entire space above the surface is divided into small elements, forming a cubic lattice such that each of the cells can be occupied by one adsorbate particle at the most, the Hamiltonian of the system can be written as [168,169]... 

While thin polymer films may be very smooth and homogeneous, the chain conformation may be largely distorted due to the influence of the interfaces. Since the size of the polymer molecules is comparable to the film thickness those effects may play a significant role with ultra-thin polymer films. Several recent theoretical treatments are available [136-144,127,128] based on Monte Carlo [137-141,127, 128], molecular dynamics [142], variable density [143], cooperative motion [144], and bond fluctuation [136] model calculations. The distortion of the chain conformation near the interface, the segment orientation distribution, end distribution etc. are calculated as a function of film thickness and distance from the surface. In the limit of two-dimensional systems chains segregate and specific power laws are predicted [136, 137]. In 2D-blends of polymers a particular microdomain morphology may be expected [139]. Experiments on polymers in this area are presently, however, not available on a molecular level. Indications of order on an... 

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