Transport modeling equations

Let us consider the selectivity parameter on the example of metal ions separation. According to the transport model equations the selectivity of two solutes, for example, two metal species, Sm,/M2> is determined by relation ... 

Examination of equation 5 shows that if there are no chemical reactions, (R = 0), or if R is linear in and uncoupled, then a set of linear, uncoupled differential equations are formed for determining poUutant concentrations. This is the basis of transport models which may be transport only or transport with linear chemistry. Transport models are suitable for studying the effects of sources of CO and primary particulates on air quaUty, but not for studying reactive pollutants such as O, NO2, HNO, and secondary organic species. 

Gaussian Plume Model. One of the most basic and widely used transport models based on equation 5 is the Gaussian plume model. 

The flow pattern is ealeulated from eonservation equations for mass and mometum, in eombination with the Algebraie Stress Model (ASM) for the turbulent Reynolds stresses, using the Fluent V3.03 solver. These equations ean be found in numerous textbooks and will not be reiterated here. Onee the flow pattern is known, the mixing and transport of ehemieal speeies ean be ealeulated from the following model equation ... 

The integration of the ventilation model into the thermal building model can be realized on different levels, from simple stack-flow equations to a full integration of a multizone airflow and contaminant transport model. 

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

In gridpoint models, transport processes such as speed and direction of wind and ocean currents, and turbulent diffusivities (see Section 4.8.1) normally have to be prescribed. Information on these physical quantities may come from observations or from other (dynamic) models, which calculate the flow patterns from basic hydrodynamic equations. Tracer transport models, in which the transport processes are prescribed in this way, are often referred to as off-line models. An on-line model, on the other hand, is one where the tracers have been incorporated directly into a d3mamic model such that the tracer concentrations and the motions are calculated simultaneously. A major advantage of an on-line model is that feedbacks of the tracer on the energy balance can be described... 

Local Average Density Model (LADM) of Transt)ort. In the spirit of the Flscher-Methfessel local average density model. Equation 4, for the pair correlation function of Inhomogeneous fluid, a local average density model (LADM) of transport coefficients has been proposed ( ) whereby the local value of the transport coefficient, X(r), Is approximated by... 

A mathematical model for reservoir souring caused by the growth of sulfate-reducing bacteria is available. The model is a one-dimensional numerical transport model based on conservation equations and includes bacterial growth rates and the effect of nutrients, water mixing, transport, and adsorption of H2S in the reservoir formation. The adsorption of H2S by the roek was considered. 

In contrast to the full equilibrium transport model, melt could be incrementally removed from the melting solid and isolated into channels for melt ascent. This model is the disequilibrium transport model of Spiegelman and Elliott (1993). Instead of substituting Equation (A7) in for Cs, the problem becomes one of separately keeping track of the concentrations of parent and daughter nuclides in the solid and the fluid. In this case, assuming steady state, two equations are used to account for the daughter nuclide ... 

Analyst to conduct multipathway human health risk assessments and food-web based ecological risk assessment modeling. BREEZE risk analyst combines databases, GIS functionality, fate, transport, and exposure modeling equations into one software application... 

In many respects, the solutions to equations 12.7.38 and 12.7.47 do not provide sufficient additional information to warrant their use in design calculations. It has been clearly demonstrated that for the fluid velocities used in industrial practice, the influence of axial dispersion of both heat and mass on the conversion achieved is negligible provided that the packing depth is in excess of 100 pellet diameters (109). Such shallow beds are only employed as the first stage of multibed adiabatic reactors. There is some question as to whether or not such short beds can be adequately described by an effective transport model. Thus for most preliminary design calculations, the simplified one-dimensional model discussed earlier is preferred. The discrepancies between model simulations and actual reactor behavior are not resolved by the inclusion of longitudinal dispersion terms. Their effects are small compared to the influence of radial gradients in temperature and composition. Consequently, for more accurate simulations, we employ a two-dimensional model (Section 12.7.2.2). 

The broadest class of models, phenomenological models, account explicitly for individual phenomena such as swelling, diffusion, and degradation by incorporation of the requisite transport, continuity, and reaction equations. This class of models is useful only if it can be accurately parameterized. As phenomena are added to the model, the number of parameters increases, hopefully improving the model s accuracy, but also requiring additional experiments to determine the additional parameters. These models are also typically characterized by implicit mean-field approximations in most cases, and model equations are usually formulated such that explicit solutions may be obtained. Examples from the literature are briefly outlined below. 

Such models are known as reactive transport models and are the subject of the next chapter (Chapter 21). We treat the preliminaries in this chapter, introducing the subjects of groundwater flow and mass transport, how flow and transport are described mathematically, and how transport can be modeled in a quantitative sense. We formalize our discussion for the most part in two dimensions, keeping in mind the equations we use can be simplified quickly to account for transport in one dimension, or generalized to three dimensions. 

Transport equations, for the surface force-pore flow model, 21 640—641 Transport gasifier, 6 798 Transport models, reverse osmosis, 21 638-639... 

This chapter is devoted to methods for describing the turbulent transport of passive scalars. The basic transport equations resulting from Reynolds averaging have been derived in earlier chapters and contain unclosed terms that must be modeled. Thus the available models for these terms are the primary focus of this chapter. However, to begin the discussion, we first review transport models based on the direct numerical simulation of the Navier-Stokes equation, and other models that do not require one-point closures. The presentation of turbulent transport models in this chapter is not intended to be comprehensive. Instead, the emphasis is on the differences between particular classes of models, and how they relate to models for turbulent reacting flow. A more detailed discussion of turbulent-flow models can be found in Pope (2000). For practical advice on choosing appropriate models for particular flows, the reader may wish to consult Wilcox (1993). 

Thus, solutions to the transported PDF equation will provide more information than is available from second-order RANS models without the problem of closing the chemical source term. 

In contrast to moment closures, the models used to close the conditional fluxes typically involve random processes. The choice of the models will directly affect the evolution of the shape of the PDF, and thus indirectly affect the moments of the PDF. For example, once closures have been selected, all one-point statistics involving U and 0 can be computed by deriving moment transport equations starting from the transported PDF equation. Thus, in Section 6.4, we will look at the relationship between (6.19) and RANS transport equations. However, we will first consider the composition PDF transport equation. 

The transported PDF equation contains more information than an RANS turbulence model, and can be used to derive the latter. We give two example derivations U) and (uuT below, but the same procedure can be carried out to find any one-point statistic of the velocity and/or composition fields.25... 

Mass transport models for multicomponent systems have been developed where the equilibrium interaction chemistry is solved independently of the mass transport equations which leads to a set of algebraic equations for the chemistry coupled to a set of differential equations for the mass transport. (Cederberg et al., 1985). 

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