Discrete phase model

Lowndes et al. [91] used the commercial CFD model Fluent to simulate flame spread along a conveyor belt. Fluent, at the time this modeling was conducted, did not contain a conventional pyrolysis model in the sense that is normally implied in the fire literature. Instead, the authors adapted a discrete phase model, which is intended to simulate the combustion of pulverized coal. 

The trajectories of spray droplets and particles and their essential parameters are calculated using the discrete phase model (DPM) formulated in Lagrange moving reference frame (the basic ideas of DPM for sprays can be found in the publications of O Rourke [26] and Crowe et al. [27]). A turbulent dispersion of the discrete phase is currently disregarded. 

S-6.2.6 Particle Tracks. Whenever the discrete phase model is used (Section 5-2.2.2), particle tracks can be used to illustrate the trajectories of the particles, bubbles, or droplets. Trajectories can usually be displayed in a number of ways. For example, lines can be colored by the time of the trajectory or temperature of the particle itself. In addition to lines, ribbons and tubes can generally be used. The tracks can be computed and displayed using the mean fluid velocities, or in the case of turbulent flows, using random fluctuations in the mean fluid velocities as well. These stochastic tracks often give a more realistic picture of the extent to which the particles reach all comers of the solution domain than do tracks computed from the mean velocities alone. 

The coal particles can be tracked as parcels in an Eulerian-Lagrangian framework. Discrete phase model (DPM) are used to define the injected particles that enter the reactor. In the case of INCI, simulation values for axial velocity of -1.732 m/s and a radial velocity of -l.Om/s of the particles must be provided. If agglomeration is neglected, a maximum particle diameter of 0.1 mm, a mean diameter of 0.09 mm, and a minimum diameter of 0.001 mm are assumed according to a Rosin-Rammler-Sperling-Bennett distribution with a spread parameter of = 0.688 in 10 individual groups for fluid-bed coal (see also Section 3.12.3.3). Particles can be treated as nonspherical with a shape factor of 0.85. 

New tools, such as the dense discrete phase models (DDPMs), could improve accuracy. 

In each of the studies described above, the reader is encouraged to watch the development of the two phases from the original homogeneous mixture. Note that there is significant clustering of like molecules well before two discrete phases form. Another observation is the ragged nature of the interface. It is not a smooth discrete separation in this molecular system level model. 

An advantage of this approach to model large-scale fluidized bed reactors is that the behavior of bubbles in fluidized beds can be readily incorporated in the force balance of the bubbles. In this respect, one can think of the rise velocity, and the tendency of rising bubbles to be drawn towards the center of the bed, from the mutual interaction of bubbles and from wall effects (Kobayashi et al., 2000). In Fig. 34, two preliminary calculations are shown for an industrial-scale gas-phase polymerization reactor, using the discrete bubble model. The geometry of the fluidized bed was 1.0 x 3.0 x 1.0 m (w x h x d). The emulsion phase has a density of 400kg/m3, and the apparent viscosity was set to 1.0 Pa s. The density of the bubble phase was 25 g/m3. The bubbles were injected via 49 nozzles positioned equally distributed in a square in the middle of the column. 

For the discrete bubble model described in Section V.C, future work will be focused on implementation of closure equations in the force balance, like empirical relations for bubble-rise velocities and the interaction between bubbles. Clearly, a more refined model for the bubble-bubble interaction, including coalescence and breakup, is required along with a more realistic description of the rheology of fluidized suspensions. Finally, the adapted model should be augmented with a thermal energy balance, and associated closures for the thermophysical properties, to study heat transport in large-scale fluidized beds, such as FCC-regenerators and PE and PP gas-phase polymerization reactors. 

A simpler version of the discrete element model is the so-called trajectory model. In the trajectory approach, droplet field is modeled as a series of trajectories that emanate from the atomizer or a starting point. The coupling effects are included by summing the heat release to and the drag force on the gas phase. This approach can be used for steady dilute flows. 

Delnoij E, Lammers FS, Kuipers JAM, van Swaaij WPM. Dynamic simulation of dispersed gas-liquid two-phase flow using a discrete bubble model. Chem Eng Sci 1997 52 1429-1458. 

The halogenation reaction of ethylene has been modeled by many researchers [170, 172-176], For chlorination in apolar solvents (or in the gas phase), the formation of two radical species requires the use of flexible CASSCF and MRCI electronic structure methods, and such calculations have been reported by Kurosaki [172], In aqueous solution, Kurosaki has used a mixed discrete-continuum model to show that the reaction proceeds through an ionic mechanism [174], The bromination reaction has also received attention [169,170], However, only very recently was a reliable theoretical study of the ionic transition state using PCM/MP2 liquid-phase optimization reported by Cammi et al. [176], These authors calculated that the free energy of activation for the ionic bromination of the ethylene in aqueous solution is 8.2 kcalmol-1, in good agreement with the experimental value of 10 kcalmol-1. 

Model. A difference equation for the material balance was obtained from a discrete reactor model which was devised by dividing the annulus into a two dimensional array of cells, each taken to be a well stirred batch reactor. The model supposes that axial motion of the mobile phase and bed rotation occur by instantaneous discontinuous jumps, between cells. Reaction occurs only on the solid surface, and for the reaction type A B + C used in this work, -dn /dt = K n - n n. Linear isotherms, n = BiC, were used, and while dispersion was not explicitly included, it could be simulated by adjusting the number of cells. The balance is given by Eq. 2, where subscript n is the cell index in the axial direction, and subscript m is the index in the circumferential direction. 

Consider an example from nucleation and growth of thin films. At least three length scales can be identified, namely, (a) the fluid phase where the continuum approximation is often valid (that may not be the case in micro- and nanodevices), (b) the intermediate scale of the fluid/film interface where a discrete, particle model may be needed, and (c) the atomistic/QM scale of relevance to surface processes. Surface processes may include adsorption, desorption, surface reaction, and surface diffusion. Aside from the disparity of length scales, the time scales of various processes differ dramatically, ranging from picosecond chemistry to seconds or hours for slow growth processes (Raimondeau and Vlachos, 2002a, b). 

Fig. 7. Schematic illustrating the coupling of a fluid-phase mass transfer model with a discrete, particle model, such as KMC, through the boundary condition. The continuum model passes the external field and the KMC simulation computes spatial and temporal rates that are needed in the boundary condition of the continuum model.

Let us first consider the catalyst/polyolefin particle in the early stage of its evolution. The particle consists of the solid catalyst carrier with catalyst sites immobilized on its surface, polymer phase, and pores. The first-principle-based meso-scopic model of particle evolution has to be capable of describing the formation of polymer at catalyst sites, transport of monomer(s) and other re-actants/diluents through the polymer and pore space, and deformation of the polymer and catalyst carrier (including its fragmentation). Similar discrete element modeling techniques have been applied previously to different problems (Heyes et al., 2004 Mikami et al., 1998 Tsuji et al., 1993). 

Indirectly related to the cell models of this section is the work of Davis and Brenner (1981) on the rheological and shear stability properties of three-phase systems, which consist of an emulsion formed from two immiscible liquid phases (one, a discrete phase wholly dispersed in the other continuous phase) together with a third, solid, particulate phase dispersed within the interior of the discontinuous liquid phase. An elementary analysis of droplet breakup modes that arise during the shear of such three-phase systems reveals that the destabilizing presence of the solid particles may allow the technological production of smaller size emulsion droplets than could otherwise be produced (at the same shear rate). 

Trapp, J. A., and Mortensen, G. A., A discrete particle model for bubble slug two-phase flow. J. Comp. Phys. 107, 367 (1993). 

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