273d

Sofiane Benyahia^{1}, **Madhava Syamlal**^{2}, and Thomas O'Brien^{2}. (1) Fluent Inc., 3647 Collins Ferry rd., Morgantown, WV 26505, (2) Simulation, Analysis & Computational, NETL/DOE, 3610, Collins Ferry Road, MS-N04, Morgantown, WV 26505

Core-annulus flow is an experimentally well established, industrially significant flow pattern of circulating fluidized beds. Several studies reported in the literature have shown that continuum gas-solids flow models are able to predict that flow pattern. But the crucial features of the model that give rise to the core-annulus flow structure have not been identified. To determine those features, we conduct transient simulations and analyze the results. Furthermore we time-average the results and investigate the formulation of time-averaged equations.

We use transient, highly resolved, 1-D, grid-independent numerical solutions of a continuum model in this study. We show that the results could be even qualitatively incorrect (high solids concentration at the center of the channel) unless grid-independence is established. This explains why in certain coarse grid computations reported in the literature it was necessary to remove a dissipation term or to increase the particle size.

Our simulations verify that the core-annulus structure arises in a time-averaged sense from unsteady gas-solids flow, as observed in experiments. We show that the key term that makes the flow unsteady is the *dissipation* term in the granular energy equation. Without that term the simulation yields a steady-state solution. The intuition based on steady-state solutions may not be valid. Unlike steady-state solutions, the transient solutions are not *unduly sensitive* to the restitution coefficient. The effect of restitution coefficient in transient simulations is remarkably different: a smaller restitution coefficient gives a higher average granular temperature.

Both the *micro-scale* (clusters resolved) and *meso-scale* (clusters time-averaged) phenomena are important, unlike turbulent single-phase flows where the meso-scale (turbulent) stresses dominate. The prediction of core-annulus flow is strongly affected by the parameters used in the (micro-scale) wall boundary conditions; it is essential that the parameters are such that no granular energy is produced at the wall. The normal stress based on kinetic theory (P_{s, micro}) is an order of magnitude larger than normal stress arising from fluctuations (P_{s, meso}). Therefore, the granular temperature and solids fraction are approximately inversely correlated, just as shown by a steady-state analysis. However, the gradient of P_{s, micro} is of the same order of magnitude as the gradient of P_{s, meso}; those gradients adjust to ensure that the time averaged total P_{s} gradient in the radial direction is zero. The meso-scale shear stress is larger than the micro-scale shear stress.

The meso-scale granular energy production term dominates the corresponding micro-scale term and must be included in time-averaged equations. That term is responsible for the maximum at the center in the granular temperature profile. The micro-scale granular energy production term is identically zero at the center because it is proportional to the gradient of solids velocity, which is zero at the center. The instantaneous gradient of solids velocity at the center, however, is not zero because of the down flow of clusters near the walls; it takes positive and negative values making the time-averaged velocity gradient exactly zero at the center. Therefore, the time-averaged square of the velocity gradient is non-zero at the center, which results in a production term in the time-averaged equations that is non-zero at the center.

We find that the predictions are insensitive to the currently available k-å type turbulence models. The traditional k-å type models, based on the experience with single phase flow calculations, may not be adequate because meso-scale terms do not necessarily dominate the micro-scale terms. And certain parameters could behave counter to our intuition based on single phase flows: we compute and confirm with physical arguments that the gas-phase turbulent (meso-scale) viscosity could become negative.

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