Clusters dense-phase

An alternate to the concept of cluster renewal discussed above is the concept of two-phase convection. This second approach disregards the separate behavior of lean and dense phases, instead models the time average heat transfer process as if it were convective from a pseudo-homogeneous particle-gas medium. Thus h hcl, hh and hd are not... 

In this model, instead of the uniform and interpenetrating continuous phases of the gas and the solids, a distinct heterogeneous structure is assumed. The elemental volume in the flow field, which has displayed observable heterogeneity, is partitioned into fractions occupied by the gas-rich, dilute phase (denoted by subscript "f") and the particle-rich, dense phase (denoted by subscript "c"), respectively. Within each "phase," uniformity is assumed, and the dense "phase" is assumed to occur as spherical clusters. That is, the dense phase is discrete, surrounded by the continuous dilute phase. In this way, eight variables... 

In this section we give examples of molecular clusters used as precursors to new dense phases or to new porous networks. 

Although particles in two-phase flow are not uniformly distributed, the dense and the dilute phases can be considered, each in its own, as uniform suspensions, and the global system can thus be regarded as consisting of dense clusters dispersed in a broth of separately distributed discrete particles, as shown in Fig. 1. The preceding correlations will therefore be used respectively in the dilute and the dense phases, for calculating micro-scale fluid-particle interaction, and also for evaluating meso-scale interphase interaction between clusters and the broth, as shown in Table I, for CD, CD[ and CD. ... 

As shown in Step 1 of Fig. 3, the fluid flowing in the dilute phase has to support the discrete particles it contains as well as the clusters in suspension, and the combined forces result in a pressure drop equal to that of the parallel dense-phase fluid flow ... 

Figure 12d shows that the dimension of clusters / is large at low gas velocities (down to dense phase continuous for bubbling fluidization), decreasing with gas velocity down to zero at the point of this sudden change, at which clusters disappear as noted already. 

In summary, for solid-type materials, molecules can form regular arrays leading to crystals, and this also may occur with colloidal-scale particles. For liquid-type materials, molecules may be dispersed on a molecular scale (i.e., dissolved) in a liquid, or they may cluster together into a separate and homogeneous phase (i.e., show phase separation). Colloidal-scale particles may also exist as separate particles in a liquid, but these particles may also cluster into a dense phase. Whether the size of the building blocks is molecular or colloidal, the phenomena of phase separation, clustering and structure formation show many similarities. 

A size distribution of particles is always desired rather than a single size in a fluidized bed. The two-phase theory of fluidized-bed operation is suspect when a bed contains appreciable lines, and models based on uniform particles should be used with caution. The dense phase in such cases should really be regarded as consisting of two phases emulsion and clusters of lines (d < 40 pm). Indeed, the results of Yadav et al. (1994) on commercial propylene ammoxidation catalyst clearly show that the lines agglomerate. A critical level of lines (30%) was found in terms of bed expansion, aeratability, and cluster size at which fluid-bed behavior is optimum. They proposed a model that takes the two dense phase components (emulsion and cluster) into account. Adding lines widens the limits of operable gas velocities and minimizes the segregation of particles. 

Yerushalmi and Cankurt considered the transport velocity as the boundary between the turbulent and the fast fluidization [4], The transition takes place gradually through a turbulent state where both voids and clusters coexist. During fast fluidization a dense phase exists at the bottom of the bed while a lean phase exists at the top. 

We often approximate the riser radial flow structure by assuming it consists of two characteristic regions a dilute gas-solid suspension preferentially traveling upward in the center (core) and a dense phase of particle clusters or strands descending near the wall (annulus), as shown in Figure 8. 

Corresponding to the bimodal distribution, at superficial gas velocity higher than the transport velocity (Grace et al, 2006), a monodisperse, air-fluidized gas—sohd two-phase flow mixture can be classified into dense clusters (denoted by subscript c) and dilute broth (denoted by subscript f) (Li and Kwauk, 1994). The dense clusters are dispersed in the continuous dilute phase. We may refine this broth-cluster structure by defining four continua or structural subelements, as shown in Fig. 6, namely, the dense-phase gas (denoted by subscript gc), the dense-phase solid (denoted by subscript pc), the dilute-phase gas (denoted by subscript gf), and the dilute-phase solid (denoted by subscript pf). The volume fraction of the dense phase is defined by f. The void fractions in the dilute and dense phases are denoted by ggf and e c, respectively. Then the sofids volume fraction is pf=l-egf in the dilute phase and pc = l- gc in the dense phase. The velocities with respect to the gas/sofids in the dilute and dense phases are tigf, tipf. tigc, and Upc, respectively. 

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... 

Here, Pc is the mixture density of the dense phase. U up i is defined by J Uf-U/), where Uf and U are mean velocities of the dilute and dense phases, respectively. This definition of mesoscale slip velocity differs a little bit from that in the cluster-based EMMS model, because the continuous phase transforms from the dilute phase to the dense phase. And their quantitative difference is l-f)PgUgc/Pc, which is normally negligible for gas-solid systems. Similarly, the closure of Fdi switches to the determination of bubble diameter. And it is well documented in literature ever since the classic work of Davidson and Harrison (1963). Compared to cluster diameter, bubble diameter arouses less disputes and hence is easier to characterize. The visual bubbles are normally irregular and in constantly dynamic transformation, which may deviate much from spherical assumption. Thus, the diameter of bubble here can also be viewed as drag-equivalent definition. 

A combination of the EMMS/matrix model for clusters and the EMMS/ bubbling model for bubbles was presented in this work. To close this set of equations, they adopted a simple criterion to quantify the phase inversion phenomenon, which is if/>l-/, then the system has dispersed bubbles in continuous dense phase, else it has continuous gas phase with dispersed clusters. Furthermore, clusters wiU exist only when input solids flux is greater than zero. The traditional stability condition of EMMS was, however, adopted to determine these structures, irrespective of bubble or cluster. 

In a gas—sohd CFB with heterogeneous reactions and mass transfer, in Hne with the structural characteristics of the SFM model (Hong et al, 2012), as shown in Fig. 12, the mass transfer and reaction in any local space can be divided into components of the dense cluster (denoted by subscript c), the dilute broth (denoted by subscript f), and in-between (denoted by subscript i), respectively. And these terms can be represented by Ri (1 = gc, gf, gi, sc, sf, si). Both the dense and dilute phases are assumed homogenous and continuous inside, and the dense phase is fiarther assumed suspended uniformly in the dilute phase in forms of clusters of particles. Then the mass transfer terms can be described with Ranz-Marshall-hke relations for uniform suspension of particles (Haider and Basu, 1988). In particular, the mesoscale interaction over the cluster will be treated as is for a big particle with hydrodynamic equivalent diameter of d. Due to dynamic nature of clusters, there are mass exchanges between the dilute and dense phases with rate ofTk (k = g, s), pointing outward from the dilute to the dense phase. 

Diffusion of gaseous reactant j through the bulk gas in the dilute phase to the surface of soHd particles in the dilute phase and the surface of clusters in the dense phase ... 

DifHision of reactant j through the surface of clusters into the bulk gas of the dense phase and to the surface of solid particles ... 

If CFB is maintained at a steady state, then the mass diffused to the particles or clusters is supposed to be equal to what is consumed in reactions. For the sake of simplicity, the reaction is assumed to be first order and happens only at the gas-solid interface. Thus, the balance equations for species concentration over the dilute and dense phases and their interface are as follows ... 

---