The Eulerian two-fluid model

When the particle Stokes number is not small, the truncated expansion for the particle velocity Up p) is no longer valid. Under these conditions, the mean particle velocity must be calculated from the disperse-phase momentum equation described in Section 4.3.7. Let us for the time being consider a very dilute population of identical particles. The mean velocity of these particles can be found by solving Eq. (4.91). For small particle 

Stokes numbers, particle trajectory crossings (PTC) are not important, in which case it is possible to neglect velocity fluctuations (i.e. upUp] 1). In this limit, the disperse-phase momenrnm balance reduces to 

If the buoyancy, gravity, drag, virmal-mass, and lift forces are accounted for, the disperse-phase momentum balance becomes (see Section 5.3.4) 

Likewise, in the monokinetic-fluid limit the fluid-phase momenrnm balance is given by 

In summary, the Eulerian two-fluid model is represented by Eqs. (5.112) and (5.113) in addition to a constitutive model for the fluid stress tensor Tf. As already mentioned, Eq. (5.112) was derived under the assumption that the particle-velocity distribution is very narrow (i.e. small particle Stokes number), and the particles must have the same internal coordinates. If these simplifications do not hold, for example under dense conditions when particle-particle collisions become important, then particle-velocity fluctuations must be taken into account, as discussed at the end of Chapter 4. 

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The Eulerian two-fluid model with particle-phase velocity that is based on the mean particle size small particle Stokes number and limited polydispersity (both in dilute and in dense systems). 

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] ... 

Eulerian two-fluid model coupled with dispersed itequations was applied to predict gas-liquid two-phase flow in cyclohexane oxidation airlift loop reactor. Simulation results have presented typical hydrodynamic characteristics, distribution of liquid velocity and gas hold-up in the riser and downcomer were presented. The draft-tube geometry not only affects the magnitude of liquid superficial velocity and gas hold-up, but also the detailed liquid velocity and gas hold-up distribution in the reactor, the final construction of the reactor lies on the industrial technical requirement. The investigation indicates that CFD of airlift reactors can be used to model, design and scale up airlift loop reactors efficiently. 

An ideal solution is to direcdy integrate the stabdity condition reflecting mesoscale mechanisms into Eulerian—Eulerian two-fluid models through the following conceptual model ... 

In the smdy by Solsvik and Jakobsen [140], a one-dimensional two-fluid model describing gas-solid flows with chemical reactions in bubbling bed reactors was derived. The reactor system was modeled in terms of mass, species mass, heat and momentum balances for each of the phases in the Eulerian reference frame. The governing equations describing the reactive flow are presented in the sequent. 

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. 

Three types of theoretical approaches can be used for modeling the gas-particles flows in the pneumatic dryers, namely Two-Fluid Theory [1], Eulerian-Granular [2] and the Discrete Element Method [3]. Traditionally the Two-Fluid Theory was used to model dilute phase flow. In this theory, the solid phase is being considering as a pseudo-fluid. It is assumed that both phases are occupying every point of the computational domain with its own volume fraction. Thus, macroscopic balance equations of mass, momentum and energy for both the gas and the solid... 

In the present study, two-dimensional Two-Fluid Eulerian model was used to describe the steady state, dilute phase flow of a wet dispersed phase (wet solid particles) in a continuous gas phase through a pneumatic dryer. The predictions of the numerical solutions were compared successfully with the results of other one-dimensional numerical solutions and experimental data of Baeyens et al. [5] and Rocha [13], Axial and the radial distributions of the characteristic properties were examined. 

Two models have been presented which are based on the Lagrangian and Eulerian difference schemes respectively. These models were able to calculate in detail the interaction of acoustic waves with inclusions in discontinuous fluids. The Lagrangian model was able to consider the transfer of normal to shear stresses at fluid discontinuities. The Eulerian model is limited to the study of dilatation waves but its computational efficiency enables it to... 

This paper presents the combined experimental/numerical investigation of the behaviour of fluid-filled plastic containers subjected to drop impact. Drop Impact experiments were conducted on original and modified bottles. During the test, strain and pressure histories were recorded at various positions. Tests were simulated numerically using the two-system FSI model. Both solid and fluid domains remain fixed during the calculations, i.e. a small-strain analysis was performed for the solid while an Eulerian fi-ame of reference was used for the fluid. This procedure was found to be simple, stable and efficient. Numerical results agreed well with experimental data, demonstrating the capability of the code to cope with this complex fluid-structure interaction problem. 

The main approach for modelling multiphase flows has been through solving conservation equations described in terms of Eulerian phase-averaged mean quantities - the two-fluid approach [665], The Eulerian mean velocity in a control volume V (such as the volume within the perimeter S of the cloud of particles) is defined as the velocity ux (for each component) averaged over the volume occupied by the fluid (ie the fluid space between the bodies),... 

The two-fluid granular flow model is formulated applying the classical Eulerian continuum concept for the continuous phase, while the governing equations of the particle phase are developed in accordance with the principles of kinetic theory. In this theory it is postulated that the particulate system can be represented considering a collection of identical, smooth, rigid spheres, adapting a Boltzmann type of equation. This microscopic balance describes the rate of change of the distribution function with respect to position and time. 

Unfortunately, the present models are still on a level aiming at reasonable solutions with several model parameters tuned to known flow fields. For predictive purposes, these models are hardly able to predict unknown flow fields with reasonable degree of accuracy. It appears that the CFD evaluations of bubble columns by use of multi-dimensional multi-fluid models still have very limited inherent capabilities to fully replace the empirical based analysis (i.e., in the framework of axial dispersion models) in use today [63]. After two decades performing fluid dynamic modeling of bubble columns, it has been realized that there is a limit for how accurate one will be able to formulate closure laws adopting the Eulerian framework. In the subsequent sections a survay of the present status on bubble column modeling is given. 

In the Lagrangian approach, the elemental control volume is considered to be moving with the fluid as a whole. In the Eulerian approach, in contrast, the control volume is assumed fixed in the space, the fluid is assumed to flow through and pass the control volume. The particle-phase equations are formulated in Lagrangian form, and the coupling between the two phases is introduced through particle sources in the Eulerian gas-phase equations. The standard k-e turbulence model, finite rate chemistry, and DTRM (discrete transfer radiation model) radiation model are used. 

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