Eulerian continuum approach

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. 

Evaluating the performance of a gas-solid transport system usually requires a means of macroscopic field description of the distribution of basic flow properties such as pressure, mass fluxes, concentrations, velocities, and temperatures of phases in the system. To conduct such an evaluation, the Eulerian continuum or multifluid approach is usually the best choice among the available approaches. 

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. 

The most fundamental step in constructing a continuum is to select a statistically valid fluid element of the continuum. The fluid element needs to be selected in such a way that it contains sufficient particles so that the fluid element possesses statistical thermodynamic properties such as temperature and density, as well as velocity. At the same time, the fluid element must be small enough compared to the characteristic length of the system so that, 

For the Eulerian continuum modeling discussed in this chapter, it is assumed that the basic form of the Navier-Stokes equation can be applied to all phases. For some dense suspension cases, the particle phase may behave as a non-Newtonian fluid. In these cases, a simple extension of the Navier-Stokes equation may not be appropriate. 

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TFM and the MP-PIC modeling that is partly continuum based. That is, subgrid modeHng is needed for both Eulerian-Eulerian and Eulerian-Lagrangian approaches. 

In the first coarse-grained approach, the discrete phase is treated as an Eulerian continuum, interpenetrating with the real continuous phase. The particle-particle interactions are then captured by an effective particle phase rheology obtained from kinetic theory of granular flows. These so-called two-fluid (Euler-Euler) models have been very successful at predicting the dynamic properties of, e.g., gas-solid fluidized beds (see Van derHoefet al, 2008 Verma et al, 2013). Despite their success, two-fluid models also have their limitations they are usually limited to idealized cases ofmonodisperse hard sphere particles, while extensions to polydisperse mixtures (e.g., in size or in contact properties) are difficult to make. Also, because no particles are explicitly tracked, it is difficult to include particle properties which may vary from particle to particle, such as particle temperature, surface moisture concentration, or chemical surface species concentrations. 

In the complete Eulerian description of multiphase flows, the dispersed phase may well be conceived as a second continuous phase that interpenetrates the real continuous phase, the carrier phase this approach is often referred to as two-fluid formulation. The resulting simultaneous presence of two continua is taken into account by their respective volume fractions. All other variables such as velocities need to be averaged, in some way, in proportion to their presence various techniques have been proposed to that purpose leading, however, to different formulations of the continuum equations. The method of ensemble averaging (based on a statistical average of individual realizations) is now generally accepted as most appropriate. 

In simulating physical operations carried out in stirred vessels, generally one has the choice between a Lagrangian approach and a Eulerian description. While the former approach is based on tracking the paths of many individual fluid elements or dispersed-phase particles, the latter exploits the continuum concept. The two approaches offer different vistas on the operations and require different computational capabilities. Which of the two approaches is most... 

Since a multiphase flow usually takes place in a confined volume, the desire to have a mathematical description based on a fixed domain renders the Eulerian method an ideal one to describe the flow field. The Eulerian approach requires that the transport quantities of all phases be continuous throughout the computational domain. As mentioned before, in reality, each phase is time-dependent and may be discretely distributed. Hence, averaging theorems need to be applied to construct a continuum for each phase so that the existing Eulerian description of a single-phase flow may be extended to a multiphase flow. 

With this approach, even the dispersed phase is treated as a continuum. All phases share the domain and may interpenetrate as they move within it. This approach is more suitable for modeling dispersed multiphase systems with a significant volume fraction of dispersed phase (> 10%). Such situations may occur in many types of reactor, for example, in fluidized bed reactors, bubble column reactors and multiphase stirred reactors. It is possible to represent coupling between different phases by developing suitable interphase transport models. It is, however, difficult to handle complex phenomena at particle level (such as change in size due to reactions/evaporation etc.) with the Eulerian-Eulerian approach. 

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... 

In accordance with the standard approach in continuum fluid mechanics, the abstract system or material control volume description is converted into a combined Eulerian framework by use of a generalization of the conventional Reynolds theorem. For general vector spaces the Re3molds theorem is written as [95, 96, 93, 94] ... 

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). 

The simulations of the gas-liquid flow are based on the Eulerian two fluid model originally derived by Ishii [21]. In this approach, each phase is treated as a continuum. After averaging the general transport equations, we get the following set of multi-phase conservation equations [19,22] ... 

One of the benefits of the Lagrangian-coordinate approach to continuum mechanics is that it provides an additional means to discover Eulerian conserved charges. 

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