Drag force closure

A graphical representation of the multilevel approach is shown in Fig. 4. All three models are now commonly accepted and are widely used by a number of research groups (both academic and industrial) around the world. In a recent paper, we have given an overview of the three models as they are employed at the University of Twente, together with some illustrative examples (Van der Hoef et al., 2004). In this chapter, we will focus on the technical details of each of the models, much of which has not been published elsewhere. The development of detailed closure relations from the simulations, as indicated in Fig. 3, is still ongoing. Some preliminary results for both the drag-force closures and solid pressure will be presented in the Sections II and III. In this chapter, we will... 

Mazzei, L. Lettieri, R 2007 A drag force closure for uniformly dispersed fluidized suspensions. Chemical Engineering Science 62, 6129-6142. 

In the case of trickle flow, it has been shown that under certain conditions the slit-flow approximation yields a very satisfactory set of constitutive equations for the gas-liquid and the liquid-solid drag forces [20, 21]. As a matter of fact, the slit flow becomes well representative of the trickle-flow regime when the liquid texture is contributed by solid-supported liquid Aims and rivulets. This generally occurs at low liquid flow rates that allow the transport of film-like liquids [20]. We will assume, without proof though, that such hypotheses also hold in the case of artificial-gravity operation. The validity of these assumptions and of the several others outlined above will be evaluated later in terms of model versus experiment comparisons. Choosing the drag force closures of the simplified Holub slit model [20], the equations system becomes ... 

The interfacial and turbulence closures suggested in the literature also differ considering the anticipated importance of the bubble size distributions. It thus seemed obvious for many researchers that further progress on the flow pattern description was difficult to obtain without a proper description of the interfacial coupling terms, and especially on the contact area or projected area for the drag forces. The bubble column research thus turned towards the development of a dynamic multi-fluid model that is extended with a population balance module for the bubble size distribution. However, the existing models are still restricted in some way or another due to the large cpu demands required by 3D multi-fluid simulations. 

For the sake of simpHcity, the mesoscale structure can be further assumed to be uniformly dispersed in forms of bubbles or clusters, both satisfying the bimodal distribution. The mesoscale drag force (Fai) can be closed with the same functional of drag but different structure properties (say, bubble diameter or cluster diameter). Accordingly, it is the closure of the mesoscale drag force that may be used to distinguish different SFMs. If the dense phase is assumed to exist in form of clusters with equivalent diameter dc, uniformly dispersed in the dilute phase, then the mesoscale drag force (Fjj) can be closed by... 

The drag coefficient is now explicitly a function of the gas fraction. Moreover, the relative velocity between the bubbly phase and the hquid phase needs to be averaged. As it is a nonlinear combination, this will, as discussed above, result in an additional term that needs to be closed. Simonin [46] derived an expression for this and provided the required closure based on mathematical derivations and some phenomenological modeling. A so-called drift velocity is introduced, which takes the nonlinear averaging into account, and the drag force (per unit volume) is written as... 

Apart from an increased drag force, high gas volume fractions can also lead to occurrence of coalescence and breakup of bubbles. Although the closures derived for these kinds of phenomena are rather mature for droplet-droplet interactions, this is not the case for bubble—bubble interactions. The main reason is probably the role of surfactants, which can have a considerable effect on the rigidness of the bubble surface and hence on the processes occurring on that scale. Given the fact that many closures were derived for water-air systems makes things worse, as the water quahty and in... 

There are three types of closure models in CFD simulation of gas—hquid flow in bubble columns, i.e., drag force, bubble-induced turbulence, and kernel functions of bubble breakup and coalescence. We will show how we utilize the EMMS approach to derive new models and integrate them into CFD simulation. 

The hierarchy of equations thereby obtained can be closed by truncating the system at some arbitrary level of approximation. The results eventually obtained by various authors depend on the implicit or explicit hypotheses made in effecting this closure—a clearly unsatisfactory state of affairs. Most contributions in this context aim at calculating the permeability (or, equivalently, the drag) of a porous medium composed of a random array of spheres. The earliest contribution here is due to Brinkman (1947), who empirically added a Darcy term to the Stokes equation in an attempt to represent the hydrodynamic effects of the porous medium. The so-called Brinkman equation thereby obtained was used to calculate the drag exerted on one sphere of the array, as if it were embedded in the porous medium continuum. Tam (1969) considered the same problem, treating the particles as point forces he further assumed, in essence, that the RHS of Eq. (5.2a) was proportional to the average velocity and hence was of the explicit form... 

Multiphase flows Eulerian-Eulerian (EE) capabilities closure/drag laws/additional forces EE-granular flows model options/Eulerian-Lagrangian (EL) true/psuedo particle model s/UD ... 

Extractables and Leachables — Materials or components derived from the container and closure that have been transferred into the contained drag substance or drug product. Forced Degradation Testing Studies [ICH Q1B] — Studies undertaken to degrade the sample deliberately. These studies, which may be undertaken in the development phase normally on the drug substances, are used to evaluate the overall photosensitivity of the material for method development purposes or degradation pathway elucidation. 

In a recent study Jakobsen et al. [71] examined the capabilities and limitations of a dynamic 2D axi-symmetric two-fluid model for simulating cylindrical bubble column reactor flows. In their in-house code all the relevant force terms consisting of the steady drag, bulk lift, added mass, turbulence dispersion and wall lift were considered. Sensitivity studies disregarding one of the secondary forces like lift, added mass and turbulent dispersion at the time in otherwise equivalent simulations were performed. Additional simulations were run with three different turbulence closures for the liquid phase, and no shear stress terms for the gas phase. A standard k — e model [95] was used to examine the effect of shear induced turbulence, case (a). In an alternative case (b), both shear- and bubble induced turbulence were accounted for by linearly superposing the turbulent viscosities obtained from the A — e model and the model of Sato and Sekoguchi [138]. A third approach, case (c), is similar to case (b) in that both shear and bubble induce turbulence contributions are considered. However, in this model formulation, case (c), the bubble induced turbulence contribution was included through an extra source term in the turbulence model equations [64, 67, 71]. The relevant theory is summarized in Sect. 8.4.4. 

At low gas volume fraction (<0.01), the forces acting on bubbles in a liquid are similar to those acting on a single bubble. Due to their low inertia as compared with the Hquid phase, bubbles are subject to a large number of forces, i.e., forces due to drag, lift, and virtual mass. In the definition of the interfacial forces, it is customary to characterize the bubble size with the sphere equivalent diameter. That is, ah effects due to nonsphericity are lumped in the closure. The closures that are most commonly applied... 

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