Multivariate Quadrature-Based Moments Methods for turbulent polydisperse gas–liquid systems

The Conditional Quadrature Method of Moments (CQMOM) and the Direct Quadrature Method of Moments (DQMOM) are compared with Direct Simulation Monte Carlo (DSMC) for the description of gas bubble coalescence, breakage and mass transfer with the surrounding continuous liquid phase. CQMOM and DQMOM are both moment methods based on the idea of overcoming the closure problem by using a quadrature approximation. The methods are compared and performances evaluated for spatially homogeneous and inhomogeneous systems. Eventually CQMOM and DQMOM are implemented in a commercial CFD code to simulate a realistic two-dimensional bubble column. Particular attention is paid to the impossibility of conserving moments with DQMOM in the presence of numerical diffusion. To cure this problem a fully-conservative DQMOM formulation is presented and tested. The relationship between the two methods is investigated, showing that under particular conditions CQMOM is identical to DQMOM. The different methods are employed under a number of different conditions including very fast chemical reactions, in order to highlight if the problem of bubble coalescence, breakage and mass transfer really needs a bivariate population balance to be tackled and what is the optimal number of nodes for the quadrature approximation.