Lagrangian trajectory model

Lagrangian trajectory models can be viewed as foUowing a column of air as it is advected in the air basin at the local wind velocity. Simultaneously, the model describes the vertical diffusion of poUutants, deposition, and emissions into the air parcel as shown in Eigure 4. The underlying equation being solved is a simplification of equation 5 ... 

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). 

Eig. 4. Schematic diagram of a Lagrangian trajectory model where Lf(/) represents the air column height in both Eulerian and Lagrangian (, Tj, ... 

Murphy, B. D., and Nelson, C. B. (1983) The treatment of ground deposition, species decay and growth and source height effects in a Lagrangian trajectory model, Atmos. Environ. 17, 2545-2547. 

After employing the preceding three assumptions, the Lagrangian trajectory model equation is simplified to... 

Pruvost J, Legrand J, Legentilhomme P, MuUer-Peuga A Lagrangian trajectory model for turbulent swirling flow in an annular cell. Comparison with RTD measurements, Chem Eng Sci 57(7) 1205-1215, 2002b. 

The second model evaluated was the Lagrangian Photochemical Model LPM (54), a trajectory model. Backward trajectories were first determined so that starting positions could be used which would allow trajectories to reach station locations at the times of measurement. Measured concentrations ranged from 0.20 to 0.26 ppm and estimated concentrations from 0.05 to 0.53 ppm. 

Another Lagrangian photochemical model was developed by Wayne et al. This photochemical model uses a moving ceir or air mass that follows an air trajectory either from a specified source or to a specified receptor. The emission input from the ground-based sources is ex-... 

FIGURE 16.20 Schematic of Lagrangian-type trajectory model (adapted from Wayne et at., 1973). 

The Eulerian continuum approach, based on a continuum assumption of phases, provides a field description of the dynamics of each phase. The Lagrangian trajectory approach, from the study of motions of individual particles, is able to yield historical trajectories of the particles. The kinetic theory modeling for interparticle collisions, extended from the kinetic theory of gases, can be applied to dense suspension systems where the transport in the particle phase is dominated by interparticle collisions. The Ergun equation provides important flow relationships, which are useful not only for packed bed systems, but also for some situations in fluidized bed systems. 

Trajectory models quantify the dynamic characteristics of particles in Lagrangian coordinates. The trajectory model is useful when the particle phase is so dilute that the description of particle behavior by continuum models may not be suitable. 

Wilson. J.D. and Sawford, B.L., 1996. Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere. Bound. Layer Meteorol., 78, pp. 191-210. 

Draxler, R.R. and Rolph, G.D., 2003. HYSPLIT (HYbrid Single-Particle Lagrangian Integrated Trajectory) Model access via NOAA ARL READY Website (http //www.arl.noaa.gov/ready/hysplit4.html). NOAA Air Resources Laboratory, Silver Spring, MD. 

In the Lagrangian frame, droplet trajectories in the spray may be calculated using Thomas 2-D equations of motion for a sphere 5791 or the simplified forms)154 1561 The gas velocity distribution in the spray can be determined by either numerical modeling or direct experimental measurements. Using the uncoupled solution approach, many CFD software packages or Navier-Stokes solvers can be used to calculate the gas velocity distribution for various process parameters and atomizer geometries/configurations. On the other hand, somesimple expressions for the gas velocity distribution can be derived from... 

Lagrangian models The next step in model development was the use of Lagrangian models depicted in Fig. f6.20 (Wayne et at., 1973). These models consider a column of air containing certain initial pollutant concentrations and follow its trajectory as it moves along a trajectory. In effect, it is an expansion of a simple box model to a series of adjacent, interconnected boxes. 

Sensitivity Studies on 1969 Trajectories with the Expanded Model. Based on the semi-Lagrangian formulation of the photochemical/diffusion model, the computed endpoint composition of the air masses depends on initial conditions, flux from the ground along the trajectories, and reaction rates. For our tests we concentrate on El Monte data because much of the polluted air there comes from somewhere else. This is believed to be a more severe test of the model than that at Huntington Park. The initial conditions are based on measurements insofar as possible. The principal initial values for the 1030 trajectory are as follows for 0730 PST (given in parts per hundred million) ... 

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