Pseudo-fluid models

Considering the variety of the pseudo-fluid models established by international colleagues, as reviewed earlier, this chapter will mainly discuss the two-phase approach, but with a brief introduction to several pseudo-fluid models developed in China. 

A k-e turbulence model was proposed by Yang (1992) to simulate the flow field in circulating fluidized beds, consisting of the following equations  

conservation for gas with respect to mass, momentum, turbulent kinetic energy and dissipation rate  

conservation for particles with respect to mass and momentum  

His model paid special attention to the influence of particles on gas turbulence and the collision between particles, and was solved numerically to calculate radial profiles of voidage, gas velocity and particle velocity, showing good agreement with experiments. 

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There are two principal approaches to formulate particle-fluid two-phase flow in fluidized beds the pseudo-fluid model and the two-phase model. 

For considering radial heterogeneity in a two-phase system, the pseudo-fluid model is simultaneously applied to the core dilute region and the wall dense region, resulting in the so-called two-channel model (Nakamura and Capes, 1973 Bai et al., 1988 Yang, 1988 Ishii et al., 1989 Berruti, 1989 Rhodes, 1990). 

Adapted from Ma X, Yuan W, Belt SEJ, James SL Better understanding of mechanochemical reactions Raman monitoring reveals surprisingly simple pseudo-fluid model for a ball milling reaction. Chem Common... 

Di Felice R. The applicability of the pseudo-fluid model to the settling velocity of a foreign particle in a suspension. Chem Eng Sci 53 371-375, 1998. 

Pseudo homogeneous models of fixed bed reactors are widely employed in reactor design calculations. Such models assume that the fluid within the volume element associated with a single catalyst pellet or group of pellets can be characterized by a given bulk temperature, pressure, and composition and that these quantities vary continuously with position in the reactor. In most industrial scale equipment, the reactor volume is so large compared to the volume of an individual pellet and the fraction of the void volume associated therewith that the assumption of continuity is reasonable. 

The One-Dimensional Pseudo Homogeneous Model of Fixed Bed Reactors. The design of tubular fixed bed catalytic reactors has generally been based on a one-dimensional model that assumes that species concentrations and fluid temperature vary only in the axial direction. Heat transfer between the reacting fluid and the reactor walls is considered by presuming that all of the resistance is contained within a very thin boundary layer next to the wall and by using a heat transfer coefficient based on the temperature difference between the fluid and the wall. Per unit area of the tube... 

Equations 1-5 completely define the "hard fluid" model for solvent induced changes in the vibrational frequency of a diatomic (or pseudo-diatomic) solute. The only adjustable parameter in this model is the coefficient Ca appearing in equation 5. The other parameters, such as the diameters of the solute and solvent as well as the solvent density and temperature, are determined using independent measurements and/or parameter correlations (37). The value of Ca can be determined with a minimal amount of experimental data. In particular we use the frequency shift observed in going from the dilute gas to a dense fluid to fix the value of Ca. Having done this, the... 

Three types of theoretical approaches can be used for modeling the gas-particles flows in the pneumatic dryers, namely Two-Fluid Theory [1], Eulerian-Granular [2] and the Discrete Element Method [3]. Traditionally the Two-Fluid Theory was used to model dilute phase flow. In this theory, the solid phase is being considering as a pseudo-fluid. It is assumed that both phases are occupying every point of the computational domain with its own volume fraction. Thus, macroscopic balance equations of mass, momentum and energy for both the gas and the solid... 

A simple model is the one-dimensional, plug-flow, pseudo-homogeneous model. In this model, we will consider the fluid and solid phases as a single phase. For this model to apply we must fulfill the following conditions ... 

The fluid-phase temperatures predicted from a two-phase pseudo-homogeneous model were shown to give reasonable agreement with experimental measurements, without explicitly adjusting the model parameters. It was demonstrated that more refined experimental measurements will be needed to determine the parameters of the model in particular, the solid and fluid phase wall Biot numbers were mutually adjustable. 

One-dimensional models basically assume that species concentrations and fluid temperature vary only in the axial direction. The only transport mechanism operating in this direction is the overall convective flow. The conservation equations may be obtained from mass and energy balance on a reference component A, over an elementary cross section of the tubular reactor. For a single reaction, the steady state conservation equations can be written for the pseudo-homogeneous model as follows ... 

As indicated by the title above we are going to do something a little new here by abandoning the strictly pseudo-homogeneous models that have been used so far, at least to the extent of considering mass transfer between the fluid and solid (adsorbent or ion-exchange resin) phases to be of importance in the overall rate process. All the systems treated will be isothermal. 

Flow of liquids or gases through fixed beds is very important in chemical reaction engineering, since many commercially important processes involve reactors that contain beds of catalyst used to promote a desired reaction. The axial dispersion model has been used extensively to model these flows, even though two phases, fluid and solid, are present. Such a pseudo-homogeneous model assumes the same form we have described in the preceding section if the Peclet number is based on particle dimension and the interstitial fluid velocity is used. In this event... 

The heat balance may be written in a similar manner, for which in our pseudo-homogeneous model the solid and fluid temperatures are taken to be the same. Thus, corresponding to we may define an average temperature t/ i-i.y s... 

Pseudo-homogeneous models. Fluid and solid phases are considered as a single pseudo-phase and the balances are imposed for only one phase. Heat and mass transport coefficients inside the bed are calculated by expressions which account for the simultaneous presence of two phases. 

In pseudo-homogeneous models it is assumed that the catalyst surface is totally exposed to the bulk fluid conditions, i.e. that there are no fluid-to-particle heat and mass transfer resistances. On the other side, heterogeneous models take conservation equations for both phases into account separately. 

E.g. in the so-called "pseudo-equilibrium model, developed by Sylvester [53-56], the same design procedure is used as in a single phase catalytic gas phase reaction, where the mass transfer resistance is replaced by a suitable overall term. Bulk flow and dispersion of the liquid phase are neglected and the whole transport mechanisms are lumped into the equilibrium of the reactant concentrations between gas-, liquid- and particle phase. It is an application of the same principle used successfully in fluid/fluid reactions [57], But the necessary precondition is that the rate of reaction is slow compared to the transfer rate across the phase boundaries, so that equilibrium can really by assured. This might be justified in some of the hydrotreating processes, but certainly not in case of an aqueous liquid phase, existing in waste water treating. Earlier models used in petroleum industry have taken in-... 

Pseudo-2D models can be especially valuable when a hierarchical strategy is employed, wherein CFD simulations are employed to obtain the transverse transport correlations that are then used in pseudo-2D models [26]. Results using this strategy for non-adiabatic microbumers are presented in subsequent sections. We use Fluent 6.2 [27] to solve a 2D eDiptic model for the combined flow, transport and reaction problem. To ensure accuracy of the Nu and Sh values computed, a non-uniform grid is chosen such that the smallest cell is 1 pm wide in the transverse direction in the fluid phase near the reactor wall. Simulations are performed for various operating conditions and Nu and Sh are computed using Equations (10.2) and (10.3). 

There are two general approaches to the prediction of the viscosity of the mixtures by the methods considered here. The first approach involves estimating the pure component viscosity of each of the constituents by some method and then combining these values to obtain the viscosity of the mixture. We refer to this approach as the multi-fluid model. A second approach is the so-called one-fluid model, in which the mixture is treated as a pseudo-pure fluid, with mixing rules for obtaining the parameters of the mixture from those of the pure components. 

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