"Spatially-Averaged Two-Fluid Models for Momentum and Heat Transport in Moderately Dense Gas-Particle Flows"
Spatially-Averaged Two-Fluid Models for Momentum and Heat Transport in Moderately Dense Gas-Particle Flows
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In gas-particle flows, clusters form in the presence of a mean body force due to the momentum coupling between the phases. Coarse computational grids are often used in the simulation of real-scale reactors in order to reduce the computation load and computation time. These coarse grids, however, do not resolve neither the micro- (individual particles, which are often three orders of magnitude smaller than the reactor dimensions) nor the meso-scale (clusters of particles, which can be up to hundreds of particle diameters large). The Spatially-Averaged Two-Fluid Model approach is aimed at predicting the influence of these unresolved meso-scale flow structures on the macro-scale flow properties in coarse-grid simulations of gas-particle flows. Thereby, spatial filters are applied to the Two-Fluid Model momentum and internal energy balance equations. The Two-Fluid Model describes both phases as continua. An additional transport equation is solved for the granular temperature, which describes the collisions and uncorrelated fluctuations of the particles. The solution of the granular temperature equation arises as a stress term in the solid phase momentum balance equation. The spatially averaging procedure allows us to describe each term in the balance equations by its mean and spatially fluctuating components. The spatially fluctuating components are those that are unresolved in coarse-grid simulations and need to be closed by valid models. The resolved momentum transfer is severely overestimated if the meso-scale structures are not accounted for, leading to completely deviant predictions for the expansion of fluidized bed reactors, the distribution of the particles and the temperature in the reactors rendering simulation results useless for the design and optimization process. We found, that the resolved interphase momentum exchange can be corrected by the drift velocity in moderately dense regimes. The drift velocity is a measure for the sub-grid heterogeneity of the flow. It can be expressed as the covariance between the gas-phase velocity and the solid volume fraction. We model it using the variances of the variables scales by a linear correlation coefficient. The correlation coefficient is estimated locally and dynamically through the application of test-filters. Transport equations are solved for the variance of the solid volume fraction and the gas-phase velocity, the turbulent kinetic energy. Due to the presence of sub-grid heterogeneities, like particle clusters, an additional turbulent kinetic energy production term arises in the transport equation. This cluster-induced turbulence production term is dependent on the drift velocity. Following a similar approach, we extend the model to include closures for the unresolved terms in the temperature balance equations. We found, that the resolved interphase heat exchange term can be corrected by a construct similar to the drift velocity, which we call drift temperature. The drift temperature is the gas-phase temperature as seen by the particles. It can be closed by the variances of the solid volume fraction and gas-phase temperature scaled by a linear correlation coefficient. Therefore, additional balance equations for the variances of the phase temperatures are derived. The additional unresolved terms in the momentum and temperature balance equations are closed based on single-phase Large-Eddy Simulation closure models.