Sankaran Sundaresan

The role of meso-scale structures in rapid gas-solid flows

Meso-scale structures that take the form of clusters and streamers are commonly observed in dilute gas–particle flows, such as those encountered in risers. Continuum equations for gas–particle flows, coupled with constitutive equations for particle-phase stress deduced from kinetic theory of granular materials, can capture the formation of such meso-scale structures. These structures arise as a result of an inertial instability associated with the relative motion between the gas and particle phases, and an instability due to damping of the fluctuating motion of particles by the interstitial fluid and inelastic collisions between particles. It is demonstrated that the meso-scale structures are too small, and hence too expensive, to be resolved completely in simulation of gas–particle flows in large process vessels. At the same time, failure to resolve completely the meso-scale structures in a simulation leads to grossly inaccurate estimates of inter-phase drag, production/dissipation of pseudo-thermal energy associated with particle fluctuations, the effective particle-phase pressure and the effective viscosities. It is established that coarse-grid simulation of gas–particle flows must include sub-grid models, to account for the effects of the unresolved meso-scale structures. An approach to developing a plausible sub-grid model is proposed.

Analysis of a frictional–kinetic model for gas–particle flow

Abstract A frictional–kinetic rheological model for dense assemblies of solids in a gas–particle mixture is described. This model treats the kinetic and frictional stresses additively. The former is modeled using the kinetic theory of granular materials. For the latter, we begin with the model described by Schaeffer [J. Differ. Equ. 66 (1987) 19] and modify it to account for strain rate fluctuations and slow relaxation of the assembly to the yield surface. Results of simulations of two model problems, namely, the gravity discharge of particles from a bin and the rise of a bubble in a fluidized bed, are presented. The simulations capture the height-independent rate of discharge of particles from the bin, the dilation of particle assembly near the exit orifice, the significant effect of the interstitial air on the discharge behavior of fine particles and the occurrence of pressure deficit above the orifice. However, the stagnant shoulder at the bottom corners of the bin is not captured; instead, one obtains a region of slow flow at the corners. The bubble rise example shows the significant effect of frictional stresses on the bubble shape. In both examples, a simplified version of the rheological model obtained by invoking a critical state hypothesis is found to be adequate.

Multiscale modeling of gas-fluidized beds

Numerical models of gas-fluidized beds have become an important tool in the design and scale up of gas-solid chemical reactors. However, a single numerical model which includes the solid-solid and solid-fluid interaction in full detail is not feasible for industrial-scale equipment, and for this reason one has to resort to a multiscale approach. The idea is that gas-solid flow is described by a hierarchy of models at different length scales, where the particle-particle and fluid-particle interactions are taken into account with different levels of detail. The results and insights obtained from the more fundamental models are used to develop closure laws to feed continuum models which can be used to compute the flow structures on a much larger (engineering) scale. Our multi-scale approach involves the lattice Boltzmann model, the discrete particle model, and the continuum model based on the kinetic theory of granular flow. In this chapter we give a detailed account of each of these models as they are employed at the University of Twente, accompanied by some illustrative computational results. Finally, we discuss two promising approaches for modeling industrial-size gas-fluidized beds, which are currently being explored independently at the Princeton University and the University of Twente.

The effect of the phase composition of model VPO catalysts for partial oxidation of n-butane

X-ray diffraction, Raman spectroscopy, 3’P MAS-NMR and spin-echo NMR indicated that model vanadium phosphorus oxide (VPO) precursors and catalysts contained various minor phases depending oxboth the synthetic approach and P/V ratios used. Raman spectroscopy revealed the presence of a number of micro-crystalline and amorphous V(W) and V(V) phases not evident by XRD. The presence of VOPO, phases was detrimental to the performance of the VP0 catalysts for KN-butane oxidation. The best model organic VP0 catalyst contained only vanadyl pyrophosphate with the highest degree of stacking order and virtually no VOPO, phase impurity. Raman spectroscopy detected vanadyl metaphosphate. VO(PO,),, in the catalysts derived from aqueous precursors possessing P/V ratios greater than I. Pure vanadyl metaphosphate catalyst was inactive in n-butane oxidation. s’P NMR demonstrated the absence of vanadyl metaphosphate and other impurity phases in the best catalyst derived from organic precursors at P/V = 1.18. The experimental data strongly indicate that the best VP0 catalysts for n-butane oxidation contain only vanadyl pyrophosphate with well-ordered stacking of the (200) planes.

Analysis of drag and virtual mass forces in bubbly suspensions using an implicit formulation of the lattice Boltzmann method

We present closures for the drag and virtual mass force terms appearing in a two-fluid model for flow of a mixture consisting of uniformly sized gas bubbles dispersed in a liquid. These closures were deduced through computational experiments performed using an implicit formulation of the lattice Boltzmann method with a BGK collision model. Unlike the explicit schemes described in the literature, this implicit implementation requires iterative calculations, which, however, are local in nature. While the computational cost per time step is modestly increased, the implicit scheme dramatically expands the parameter space in multiphase flow calculations which can be simulated economically. The closure relations obtained in our study are limited to a regular array of uniformly sized bubbles and were obtained by simulating the rise behaviour of a single bubble in a periodic box. The effect of volume fraction on the rise characteristics was probed by changing the size of the box relative to that of the bubble. While spherical bubbles exhibited the expected hindered rise behaviour, highly distorted bubbles tended to rise cooperatively. The closure for the drag force, obtained in our study through computational experiments, captured both hindered and cooperative rise. A simple model for the virtual mass coefficient, applicable to both spherical and distorted bubbles, was also obtained by fitting simulation results. The virtual mass coefficient for isolated bubbles could be correlated with the aspect ratio of the bubbles.

Instabilities and the formation of bubbles in fluidized beds

As is well known, most gas-fluidized beds of solid particles bubble; that is, they are traversed by rising regions containing few particles. Most liquid-fluidized beds, on the other hand, do not. The aim of the present paper is to investigate whether this distinction can be accounted for by certain equations of motion which have commonly been used to describe both types of bed. For the particular case of a bed of 200 μm diameter glass beads fluidized by air at ambient conditions it is demonstrated, by direct numerical integration, that small perturbations of the uniform bed grow into structures resembling the bubbles observed in practice. When analogous computations are performed for a water-fluidized bed of 1 mm diameter glass beads, using the same equations, with parameters modified only to account for the greater density and viscosity of water and to secure the same bed expansion at minimum fluidization, it is found that bubble-like structures cannot be grown. The reasons for this difference in behaviour are discussed.

Bridging the rheology of granular flows in three regimes

We investigate the rheology of granular materials via molecular dynamics simulations of homogeneous, simple shear flows of soft, frictional, noncohesive spheres. In agreement with previous results for frictionless particles, we observe three flow regimes existing in different domains of particle volume fraction and shear rate, with all stress data collapsing upon scaling by powers of the distance to the jamming point. Though this jamming point is a function of the interparticle friction coefficient, the relation between pressure and strain rate at this point is found to be independent of friction. We also propose a rheological model that blends the asymptotic relations in each regime to obtain a general description for these flows. Finally, we show that departure from inertial number scalings is a direct result of particle softness, with a dimensionless shear rate characterizing the transition.

Evolution of the active surface of the vanadyl pyrophosphate catalysts

Bulk crystallinity of vanadyl(IV) pyrophosphate catalysts forn-butane partial oxidation increased up to 23 days on stream as determined by XRD and Raman spectroscopy, while selectivity reached steady state after 8–10 days. Electron microscopy detected a 15 Å amorphous layer terminating the (200) planes of (VO)2P2O7 in fresh catalysts that was not observed in the equilibrated catalysts. It is suggested that ordering of (200) planes at the surface of (VO)2P2O7 is responsible for selective oxidation.

Electrical capacitance tomography measurements on vertical and inclined pneumatic conveying of granular solids

Abstract Pneumatic conveying of granular solids in vertical and inclined risers was studied using electrical capacitance tomography (ECT). The focus of the study was on flow development past a smooth bend connecting the riser to a horizontal duct which brought the gas-particle mixture to the riser. In the vertical riser, dispersed flow manifested a core–annular structure, whose development is discussed. Three different time-dependent flow patterns were imaged. Slugging flow, which appeared to be intrinsic to riser flow, took the form of alternating bands of core–annular disperse flow and a slug with a particle-rich core. Averaging over these two structures yielded a composite distribution with high particle concentration both at the axis and the wall region. Pulsing flow, whose ECT fingerprint was similar to that of slugging flow, was largely an entrance effect. Stationary and moving annular capsules with a dilute core were also observed, and such flow patterns do not appear to have been reported previously. Our ECT measurements probing the development of disperse flow in an inclined riser past a bend revealed that the particle loading initially decreased, subsequently increased and then leveled off. Regimes such as eroding dune flow and flow over a settled layer could be easily imaged using ECT. The surface of the settled layer had a concave shape, suggesting that the particles were picked up from the settled layer by airflow at the center and deposited on the sides of the tube.

Direct numerical simulations of dense suspensions: Wave instabilities in liquid-fluidized beds

We present results of direct numerical simulations of travelling waves in dense assemblies of monodisperse spherical particles fluidized by a liquid. The cases we study have been derived from the experimental work of others. In these simulations, the flow of interstitial fluid is solved by the lattice-Boltzmann method (LBM) and the particles move under the influence of gravity, hydrodynamic forces stemming from the LBM, subgrid-scale lubrication forces and hard-sphere collisions. We first show that the propagating inhomogeneous structures seen in the simulations are in agreement with those observed experimentally. We then use the detailed information contained in the simulation results to assess aspects of two-fluid model closures, namely, fluid–particle drag, and the various contributions to the effective stresses. We show that the rates of compaction and dilation of the particle phase in the travelling waves are comparable to the rate at which the microstructure relaxes, and that there is a pronounced effect of the rate of compaction on the average collisional normal stress. Although this effect can be expressed as an effective bulk viscosity term, this approach would require the use of a path-dependent bulk viscosity. We also find that the effective fluid–particle drag coefficient can be described well with the often-used closure motivated by the experiments of Richardson & Zaki (Trans. Inst. Chem. Engng vol. 32, 1954, p. 35). In this respect, the effect of the system size for determining the drag requires specific care. The shear viscosity of the particle phase manifests small, but clearly noticeable dependence on the rate of compaction/dilation of the particle phase. Our observations point to the need for higher-order closures that recognize the slow evolution of the microstructure in these flows and account for the effects of non-equilibrium microstructure on the stresses.