Lagrangian Trajectory Approach

Typical Lagrangian approaches include the deterministic trajectory method and the stochastic trajectory method. The deterministic trajectory method neglects all the turbulent transport processes of the particle phase, while the stochastic trajectory method takes into account the effect of gas turbulence on the particle motion by considering the instantaneous gas velocity in the formulation of the equation of motion of particles. To obtain the statistical 

The development of the Lagrangian models has been limited mainly by the inherent need for large computing capacity to carry out statistical averaging and computation of the phase interactions. The Lagrangian models are particularly applicable to very dilute or discrete flow situations for which multifluid models are not appropriate, or to situations in which the historic tracking of particles is important (such as in pulverized coal combustion in a furnace or the tracking of radioactive particles in gas-solid flows). 

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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. 

For example, assume that our goal is to simulate air quality in a given location, say, Claremont, California on a given day, for example, August 28,1987. The first step using a Lagrangian modeling approach is to calculate, based on the wind field, the paths for the 24 air parcels that arrived at Claremont at 0 00,1 00,..., and 23 00 of the specific date. These backward trajectories can be calculated if we know the three-dimensional wind field ( , ) ,z,t),uy(x,y,z,t), and uz(x,y,z,t) from... 

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

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... 

In actual applications, the gas flow in a gravity settler is often nonuniform and turbulent the particles are polydispersed and the flow is beyond the Stokes regime. In this case, the particle settling behavior and hence the collection efficiency can be described by using the basic equations introduced in Chapter 5, which need to be solved numerically. One common approach is to use the Eulerian method to represent the gas flow and the Lagrangian method to characterize the particle trajectories. The random variations in the gas velocity due to turbulent fluctuations and the initial entering locations and sizes of the particles can be accounted for by using the Monte Carlo simulation. Examples of this approach were provided by Theodore and Buonicore (1976). 

Lagrangian, respectively. At first we note that the total energy is conserved in both the dynamics, with oscillations orders of magnitude smaller than the oscillations of the potential energy. The latter presents on the other hand a behavior that is quite different in the two cases. For the case in which the charges are equilibrated at each step, the oscillations are quite large, of the order of 3.5 x 10-3 au, and they last for the whole trajectory. On the other hand, for the extended Lagrangian approach, after an initial period... 

With a Eulerian-Lagrangian approach, processes occurring at the particle surface can be modeled when simulating particle trajectories (for example, the process of dissolution or evaporation can be simulated). However, as the volume fraction of dispersed phase increases, the Eulerian-Lagrangian approach becomes increasingly computation intensive. A Eulerian-Eulerian approach more efficiently simulate such dispersed multiphase flows. 

In the second part, flow in the vapor space of the separator, where the gas phase is a continuous phase, was modeled. An Eulerian-Lagrangian approach was used to simulate trajectories of the liquid droplets since the volume fraction of the dispersed liquid phase is quite small. The grid used for the vapor space is shown in Fig. 9.20. The simulated gas volume fraction distribution near the gas-liquid interface and corresponding gas flow in the vapor space are shown in Fig. 9.22. The gas volume fraction distribution and the gas velocity obtained from the model of the bottom portion of the loop reactor were used to specify boundary conditions for the vapor space model. In addition to the gas escaping from the gas-liquid interface, it is necessary to estimate the amount of liquid thrown into the vapor space by the vapor bubbles erupting at the... 

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. 

There are two main approaches for the numerical simulation of the gas-solid flow 1) Eulerian framework for the gas phase and Lagrangian framework for the dispersed phase (E-L) and 2) Eulerian framework for all phases (E-E). In the E-L approach, trajectories of dispersed phase particles are calculated by solving Newton s second law of motion for each dispersed particle, and the motion of the continuous phase (gas phase) is modeled using an Eulerian framework with the coupling of the particle-gas interaction force. This approach is also referred to as the distinct element method or discrete particle method when applied to a granular system. The fluid forces acting upon particles would include the drag force, lift force, virtual mass force, and Basset history force.Moreover, particle-wall and particle-particle collision models (such as hard sphere model, soft sphere model, or Monte Carlo techniques) are commonly employed for this approach. In the E-E approach, the particle cloud is treated as a continuum. Local mean... 

The question of what controls the asymptotic decay rate and how is it related to characteristic properties of the velocity field has been an area of active research recently, and uncovered the existence of two possible mechanisms leading to different estimates of the decay rate. Each of these can be dominant depending on the particular system. One theoretical approach focuses on the small scale structure of the concentration field, and relates it to the Lagrangian stretching histories encountered along the trajectories of the fluid parcels. This leads to an estimate of the decay rate based on the distribution of finite-time Lyapunov exponents of the chaotic advection. Details of this type of description can be found in Antonsen et al. (1996) Balkovsky and Fouxon (1999) Thiffeault (2008). Here we give a simplified version of this approach in term of the filament model based... 

The Lagrangian approach to turbulent diffusion is concerned with the behavior of representative fluid particles. We therefore begin by considering a single particle that is at location x at time t in a turbulent fluid. The subsequent motion of the particle can be described by its trajectory, X[x, r], that is, its position at any later time t. Let... 

Crowe et al. (1977) proposed an axi-symmetric spray drying model called Particle-Sonrce-In-Cell model (PSI-Cell model). This model includes two-way mass, momentum, and thermal conpling. In this model, the gas phase is regarded as a continuum (Eulerian approach) and is described by pressure, velocity, temperature, and humidity fields. The droplets or particles are treated as discrete phases which are characterized by velocity, temperature, composition, and the size along trajectories (Lagrangian approach). The model incorporates a finite difference scheme for both the continuum and discrete phases. The authors used this PSI-Cell model to simulate a cocurrent spray dryer. But no experimental data were compared with it. More details can be found in the woik by Crowe et al. (1977). 

Based on the solution obtained for the flow field of the continuous phase, using an Enler-Lagrangian approach, we can obtain the particle trajectories by solving the force balance for the particles taking into account the discrete phase inertia, aerodynamic drag, gravity g, and further optional user-defined forces... 

Particle trajectory is the result of the interaction of the particle with the electric field and the flow field. To simulate the particle trajectories, there are two approaches. The first approach is the Lagrangian tracking method, which neglects the finite size of the particles and treats them as point particles and solves the field variables without the presence of the particles [8]. In this case, only the effect of the field variables on the particle is considered. The second approach is the stress tensor approach, which includes the size effect of the particle. In this approach, the field variables are solved with the presence of the finitesized particle, and the particle translates as a result of the interaction of the particle with the electric and flow field [8]. In each incremental movement of the particle, the field variables need to be resolved. The former approach is very simple and works good to some extent, and the latter approach is accurate yet computationally expensive. 

Semi-Lagrangian method Another numerical time integration approach based on the Lagrangian method, which calcrrlates the properties of fluid following the fluid parcels. To avoid the distortion of parcel trajectories, new fltrid parcels ate selected at legrtlarly distributed grid points at each time step. 

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