Eulerian Modeling

In Eulerian particle modeling, the particles are considered as a continuous fluid, just like the gas. This fluid interpenetrates with the gas in the cyclone, and interacts with it, in accordance with the known laws of interaction, for instance, Stokes law. Transport equations, which are coupled through the interaction terms, are solved for both fluid and particle phases. 

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Trajectory models require spatiaUy and temporaUy resolved wind fields, mixing-height fields, deposition parameters, and data on the spatial distribution of emissions. Lagrangian trajectory models assume that vertical wind shear and horizontal diffusion are negligible. Other limitations of trajectory and Eulerian models have been discussed (30). 

Euleria.n Models. Of the Eulerian models, the box model is the easiest to conceptualize. The atmosphere over the modeling region is envisioned as a well-mixed box, and the evolution of pollutants in the box is calculated following conservation-of-mass principles including emissions, deposition, chemical reactions, and atmospheric mixing. 

Hansen, D. A., Puckett, K. ]., Jansen, ]. ]., Lusis, M., and Vickery, J. S., The Eulerian Model Evaluation Field Study (EMEFS). Paper 5.1, pp 58-62, in "Preprints, Seventh Joint Conference on Applications of Air Pollution Meteorology with AWMA," Jan. 14-18, 1991, New Orleans. American Meteorological Society, Boston, 1991. 

Pai P, Karamchandani P, Seigneur C. 1997. Simulation of the regional atmospheric transport and fate of mercury using a comprehensive Eulerian model. Atmos Environ 31 2717-2732. 

The methods used for modeling pure granular flow are essentially borrowed from that of a molecular gas. Similarly, there are two main types of models the continuous (Eulerian) models (Dufty, 2000) and discrete particle (Lagrangian) models (Herrmann and Luding, 1998 Luding, 1998 Walton, 2004). The continuum models are developed for large-scale simulations, where the controlling equations resemble the Navier-Stokes equations for an ordinary gas flow. The discrete particle models (DPMs) are typically used in small-scale simulations or... 

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A number of different approaches have been taken to understanding the VOC-NO chemistry and its application to control strategy developments. These include the use of environmental chambers, models ranging from simple linear rollback to complex Eulerian models, and field studies. 

Models currently in use for developing control strategy options are grid-based, or Eulerian, models, the... 

For example, Kumar and Russell (1996) examined the effect on predicted ozone levels in the Los Angeles area of two different approaches now used for incorporating meteorology into a grid-based Eulerian model. The diagnostic approach is based on field measurements of the needed meteorological variables and in-... 

Figure 16.26, for example, shows ozone isopleths predicted for the Los Angeles area using the CIT Eulerian model under two different assumptions for the boundary and initial conditions (Winner et al., 1995). In Fig. 16.26a, the pollutant concentrations measured in the past were used to set the boundary and initial conditions. In Fig. 16.26b, concentrations measured in clean air over the Pacific Ocean upwind of the modeled area were used instead. The axes are given in terms of the percentage of VOC or NOx emissions compared to the baseline level on August 28, 1987, rather than in terms of absolute concentrations as for... 

Similarly, increasing the on-road exhaust vehicle emissions by a factor of three in a Eulerian model was found to improve the agreement between predicted and observed concentrations of O-, and organics in the... 

In the present study, two-dimensional Two-Fluid Eulerian model was used to describe the steady state, dilute phase flow of a wet dispersed phase (wet solid particles) in a continuous gas phase through a pneumatic dryer. The predictions of the numerical solutions were compared successfully with the results of other one-dimensional numerical solutions and experimental data of Baeyens et al. [5] and Rocha [13], Axial and the radial distributions of the characteristic properties were examined. 

Sportisse B. (2000). Box models versus Eulerian models in air pollution modelling. Atmospheric Environment, 35(1), 173-178. 

Durham, J. Dennis, R. Laulainen, N. Renne, D. Pennell, B. Barchett, R. Hales, J. "Regional Eulerian Model Field Study and Evaluations Proposed Management and Technical Approaches," EPA Office of Research and Development, Aug., 1986. 

Two models have been presented which are based on the Lagrangian and Eulerian difference schemes respectively. These models were able to calculate in detail the interaction of acoustic waves with inclusions in discontinuous fluids. The Lagrangian model was able to consider the transfer of normal to shear stresses at fluid discontinuities. The Eulerian model is limited to the study of dilatation waves but its computational efficiency enables it to... 

Gillani N. V. and Pleim J. E. (1996) Sub-grid-scale features of anthropogenic emissions of NO. and VOC in the context of regional Eulerian models. Atmos. Environ. 30, 2043 —2059. 

In general, it may be concluded that it is possible to develop appropriate Eulerian-Eulerian models to simulate complex gas-liquid (solid) flows, with some support from the experimental data. Some of the possible applications of such models are discussed in the next section. Before discussing these applications, recent simulations carried out with Eulerian-Lagrangian and volume of fluid (VOF) approaches are briefly reviewed here. 

Pfleger, D., Gomes, S., Gilbert, N. and Wagner, H.-G. (1999), Hydrodynamic simulations of laboratory scale bubble columns fundamental studies of the Eulerian-Eulerian modeling approach, Chem. Eng. Sci., 54, 5091-5099. 

Eulerian-Eulerian models Simulations of complex industrial... 

Other regional transport models, such as the Regional Lagrangian Model of Air Pollution (RELMAP Eder et al., 1986), use a different computational scheme than Eulerian models. In a Lagrangian model, the coordinate system moves with a parcel of air and mass balance of pollutant concentrations is computed on a parcel as it moves through space. 

Each droplet draws its own trajectory in space that depends on its radius r. One could easily get a Lagrangian model for the droplet ensemble if the air velocity field would be exactly known. The problem consists however in the fact that the latter field V(x, z) itself is dependent on all the moving obstructions and their complex velocity field V(x,y z). Interpreting droplets as a continuous medium, as a gas of droplets , and substituting the individual derivatives by local ones, one gets the following Eulerian model for it ... 

A combination of the forward and inverse modelling approaches allows to solve some environmental and nuclear risk problems more effectively compared with the traditional ways based on the forward modelling. For the inverse modelling problem, most of the western scientists (Persson et al., 1987 [491] Prahm et al., 1980 [509] Seibert, 2001 [569]) use the common back- trajectory techniques, suitable only for the Lagrangian models. The Novosibirsk scientific school established by G.I. Marchuk in Russia has suggested a fruitful theoretical method for inverse modelling, based on adjoint equations (Marchuk, 1982 [391], 1995 [392] Penenko, 1981 [486]) and suitable for the Eulerian models. This approach has further been used and improved by several authors (Baklanov, 1986 [20], 2000 [25] Pudykiewicz, 1998 [512] Robertson and Lange, 1998 [538]) for estimation of source-term parameters in the atmospheric pollution problems. 

In this section the governing equations determining the averaged Eulerian-Lagrangian and the Eulerian - Eulerian modeling concepts are discussed. 

Mathiesen V, Solberg T, Hjertager BH (2000) Predictions of gas/particle flow with an Eulerian model including a reahstic particle size distribution. Powder Technology 112 34-45... 

For applications in Eulerian model formulations the particles are assumed small and the dispersion is sufficiently dilute so that there is no significant particle-particle interactions [64] [65]. The net force on the particles per unit of mixture volume, Fp, is then obtained by multiplying fp by the number density of particles, Np (e.g., Albraten [1]) ... 

Auton [7], Thomas et al [152] and Auton et al [8] determined a lift force due to inviscid flow around a sphere. In an Eulerian model formulation this lift force parameterization is usually approximated for dilute suspensions, giving ... 

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