Simulations, binary collision model

Even though the binary collision model is very useful, it is still an approximation to the real situation and a more detailed understanding probably requires computer simulation studies such as those pioneered by Harrison . 

Simulations. In addition to analytical approaches to describe ion—solid interactions two different types of computer simulations are used Monte Carlo (MC) and molecular dynamics (MD). The Monte Carlo method relies on a binary collision model and molecular dynamics solves the many-body problem of Newtonian mechanics for many interacting particles. As the name Monte Carlo suggests, the results require averaging over many simulated particle trajectories. A review of the computer simulation of ion—solid interactions has been provided (43). 

Computer Simulations Based on the Binary Collision Model. 351... 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. 

In a hard sphere approach, particles are assumed to interact through instantaneous binary collisions. This means particle interaction times are much smaller than the free flight time and therefore, hard particle simulations are event (collision) driven. For a comprehensive introduction to this type of simulation, the reader is referred to Allen and Tildesley (1990). Hoomans (2000) used this approach to simulate gas-solid flows in dense as well as fast-fluidized beds. There are three key parameters in such hard sphere models, namely coefficient of restitution, coefficient of dynamic friction and coefficient of tangential restitution. Coefficient of restitution is discussed later in this chapter. Detailed discussion of these three model parameters can be found in Hoomans (2000). 

The binary collision approximation (BCA) model was the first to be used in computer simulations of ion-solid interactions (Bredov et al, 1958). The usefulness of computer simulations was further demonstrated by Robinson and Oen (1963) during their discovery of the channeling effect. Computer simulations based on the BCA model in essence fall into two categories, those that assume a crystalline structure of the solid and those that, as in calculations based on the TRIM code, assume a randomized or structureless target. 

The validity of the viscoelastic model (5.32) has been tested against experimental and molecular dynamics simulation results [26, 27, 28]. The detailed comparison has established that the viscoelastic model works remarkably well for wavenumbers k km, where km denotes the first peak position of the static structure factor S k). However, it has also been found that the situation is not so satisfactory for smaller wavenumbers, where the viscoelastic model is shown in some circumstances to yield even qualitatively incorrect results. This failure was attributed to the fact that the single relaxation time model (5.31) cannot describe both the short-time behavior of the memory function, dominated by the so-called binary collisions, and in particular the intermediate and long-time behavior where in the liquid range additional slow processes play an important role (see the next subsection). It is obvious that these conclusions demand a more rigorous consideration of the memory function, which lead to the development of the modern version of the kinetic theory. Nevertheless, the viscoelastic model provides a rather satisfactory account of the main features of microscopic collective density fluctuations in simple liquids at relatively large wavenumbers, and its value should not be undervalued. 

Abstract We put together the state of knowledge on binary colUsional interactions of droplets in a gaseous environment. Phenomena observed experimentally after drop collisions, such as coalescence, bouncing, reflexive separation and stretching separation, are discussed. Collisions of drops of the same liquid and of different -miscible or immiscible - liquids, as well as collisions of drops of equal and different size are addressed. Collisions of drops of immiscible liquids may lead to an unstable interaction which is not observed with drops of equal or miscible liquids. Regimes characterized by the various phenomena are depicted in nomograms of the Weber number and the non-dimensional impact parameter. The state-of-the-art in the simulation of binary droplet collisions is reviewed. Overall three different methods are represented in the literature on these simulations. We discuss models derived from numerical simulations and from experiments, which are presently in use for simulations of spray flows to account for the influence of coUisional interactions of the spray droplets on the drop size spectrum of the spray. 

Keywords Binary drop colUsions Bouncing Coalescence Collision model Crossing separation Gaseous environment Immiscible liquids Lattice-Boltzmann simulation Miscible liquids Navier-Stokes simulation Reflexive separation Satellite droplets Spray flow simulation SPH simulation Stretching separation... 

Particle collision is important in simulating bubble behavior in liquid-solid suspensions, especially for high solids holdup conditions. A simulation without considering particle collision leads to inappropriate nonuniformity of particle distribution in the flow field and hence false flow field information. Numerical instability may also occur as a result of inappropriate particle accumulation in a computational cell. In the collision model discussed above, only the binary-particle collision mechanism is considered, which limits the model to low solids holdup conditions (less than 30-40% by volume). For higher solids holdup cases, the multiparticle collision mechanism needs to be considered. 

Two of the most common classes of particle-dynamic simulations are termed hard-particle and soft-particle methods. Hard-particle methods calculate particle trajectories in response to instantaneous, binary collisions between particles, and allow particles to follow ballistic trajectories between collisions. This class of simulation permits only instantaneous contacts and is consequently often used in rapid flow situations such as are found in chutes, fluidized beds, and energetically agitated systems. Soft-particle methods, on the other hand, allow each particle to deform elastoplastically and compute responses using standard models from elasticity and tribology theory. This approach permits enduring particle contacts and is therefore the method of choice for mmbler apphcations. The simulations described in this chapter use soft-particle methods and have been validated and found to agree in detail with experiments. 

RMD Simulation of Chemical Nucleation (22). A series of microscopic computer experiments was performed using the cooperative isomerization model (Eq. 2). This system was selected for the trial simulations for several reasons First, only two chemical species are involved, so that a minimal number of particles is needed. Second, the absence of buffered chemicals (e.g., A and B in the Trimolecular reaction of the next section) eliminates the need for creation or destruction of particles in order to maintain constant populations (19., 22j. Third, the dynamical model of the cooperative mean-field interaction can be examined as a convenient means of introducing cubic or higher nonlinearity into molecular models based on binary collisions. Finally, the need for a microscopic simulation is most apparent for transitions between multi -pie macroscopic states. Indeed, the characterization of spatially localized fluctuations is of obvious importance to the understanding of nucleation phenomena. As for the equilibrium vapor-liquid and liquid-solid transitions, detailed simulations at the molecular level should provide deep physical insight into chemical nucleation processes whkh is unattainable from theory, higher-level simulation, or experiment. 

The present chapter puts together the state of knowledge about the coUisional interaction of liquid droplets and gives an overview of the literature on experiments, as well as on simulation and modeling of binary liquid droplet collisions, both as an elementary phenomenon and also in the context of spray flows. 

Modeling Binary Droplet Collisions in Spray Simulations... 

Modeling of the complicated phenomena in binary droplet collisions occurring in spray flows is difficult due to the variety of potential outcomes from a collision [69-71]. The first necessity is to predict the stability against stretching or reflexive separation. Then, for unstable drop collisions, the resulting drop sizes need to be predicted. All predictions should follow from algebraic models without the need to solve additional transport equations in the spray flow code to account for the collisions. Needless to say that it is impossible to simulate the full detail of the processes in droplet collisions, as done in the simulations discussed in section Simulations of Droplet Collisions, in the course of a spray flow simulation [72-82]. 

---