Trajectory modeling of multiphase flows

The boundary conditions for the gas phase are assumed to be of the same form as that for the single gas phase discussed in 5.2.5. For the particle phase, the boundary conditions are given as follows. 

For the normal components of particle velocities, the boundary conditions may be expressed by using the coefficient of restitution as 

The boundary conditions of particle temperature may be obtained from the heat transfer due to the collision of two bodies. Sun and Chen (1988) formulated the heat transfer per impact of two elastic particles by considering the collision of two elastic particles with different temperatures and assuming the heat conduction occurs only in the normal direction. 

We may consider the solid wall as a very large particle with an infinite thermal capacity. Thus, for the collision between a particle and a solid wall, we have 

For the boundary condition of particle concentration at the wall, a zero normal gradient condition is frequently adopted that is 

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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. 

As it will be briefly described in this section, the phenomenology of droplet flows in microfluidic networks can be effectively recovered by simple numerical models [39, 41-43] that assume the generic features of the multiphase microfluidic flows. These very basic models offer an insight into the qualitative features of the dynamics. As the understanding and characterization of the flow of droplets and bubbles in capillaries, junctions, comers etc. progresses, it can be expected that the same simple models, with the appropriate numeric input will able to predict the trajectories of droplets flowing in real microfluidic networks. At the end of this section we provide an example of a quantitative match between experiment and numerical simulation. 

Inhomogeneous or multiphase reaction systems are characterised by the presence of macroscopic (in relation to the molecular level) inhomogeneities. Numerical calculations of the hydrodynamics of such flows are extremely complicated. There are two opposite approaches to their characterisation [63, 64] the Euler approach, with consideration of the interfacial interaction (interpenetrating continuums model) and the Lagrange approach, of integration by discrete particle trajectories (droplets, bubbles, and so on). The presence of a substantial amount of discrete particles in real systems makes the Lagrange approach inapplicable to study motion in multicomponent systems. Under the Euler approach, a two-phase flow is described... 

Trajectories and singular points in steady-state models of two-phase flows. Int. J. Multiphase Flow 511-533. 

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