Continuum modeling of multiphase flows

Euler-Euler models assume interpenetrating continua to derive averaged continuum equations for both phases. The probability that a phase exists at a certain position at a certain time is given by a phase indicator function, which, for steady-state processes, is equivalent to the volume of fraction of the correspondent phase (volume-of-fluid technique). The phase-averaging process introduces further unknowns into the basic conservation equations their description requires empirical and problem-dependent input (94). In principal, Euler-Euler models are applicable to all multiphase flows. Advantages and disadvantages of both methods are compared, e.g., in Refs. 95 and 96. 

The Eulerian continuum approach is basically an extension of the mathematical formulation of the fluid dynamics for a single phase to a multiphase. However, since neither the fluid phase nor the particle phase is actually continuous throughout the system at any moment, ways to construct a continuum of each phase have to be established. The transport properties of each pseudocontinuous phase, or the turbulence models of each phase in the case of turbulent gas-solid flows, need to be determined. In addition, the phase interactions must be expressed in continuous forms. 

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... 

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