Quadrature-based moment methods for the simulation of fluid-fluid multiphase systems

Fluid-fluid multiphase systems are complicated by the fact that the disperse phase is constituted by deformable bubbles or droplets, that have a high tendency to coalesce and break, generating complex phase coupling for mass, heat and momentum. Different techniques can be used to simulate these systems, but Eulerian approaches are nowadays the ones directly applicable to industrial problems. Historically the problem was studied with simplified assumptions for the fluid dynamics, and focusing of bubble/droplet coalescence and break up, or viceversa, by assuming constant size and composition for bubbles/droplets and investigating the fluid dynamics. This has changed since the introduction of quadrature-based moment methods (QBMM), where the state of the poly-disperse population of bubbles/droplets is tracked by solving a specific equation (called generalized population balance equation) that dictates the evolution of a number density function (NDF). The above mentioned methods are based on the simple idea of solving transport equations for the moments of this NDF and by overcoming the closure problem by adopting a quadrature approximation. In this talk three of these methods are discussed in details: the Quadrature Method of Moments (QMOM), the Direct Quadrature Method of Moments (DQMOM) and the Conditional Quadrature Method of Moments (CQMOM). Particular attention is paid to the specific theoretical and numerical issues encountered in the simulation of gas-liquid systems. The talk will focus on the problem of correctly predicting momentum and mass phase coupling, as well as bubble coalescence and break up. Numerical issues related to spatial discretization, to the use of conservative (i.e., moments) versus primitive (i.e., quadrature nodes and weights) and to the problem of moment realizability, will also be thoroughly analyzed. Moreover, by discussing different systems operating under different operating conditions, issues related to the number of internal coordinates to be tracked and the number of nodes of the quadrature approximation to be used, will be discussed and addressed. Comparison of QBMM with other more sophisticated techniques, such as Direct Simulation Monte Carlo (DSMC), will also be discussed and eventually some guidelines for the use of QBMM will be presented. The talk will conclude with the presentation of some industrially relevant case studies conducted in stirred tanks and bubble columns, together with their validation with experiments in terms of bubble size distributions and mass transfer rates. Finally the main limitations of these methods and the need of better physics for specific interfacial phenomena (accounting for example for the presence of surfactants) will be discussed.