Evaluation of Weighted Residual Methods for the Solution of a Population Balance Model Describing Bubbly Flows: The Least-Squares, Galerkin, Tau, and Orthogonal Collocation Methods

In dispersed gas–liquid flows, the bubble size distribution plays an important role in the phase structure and interphase forces, which, in turn, determine the multiphase hydrodynamic behaviors, including the spatial profiles of the gas fraction, gas and liquid velocities, and mixing and mass-transfer behaviors. Thus, fluid particle coalescence and breakage phenomena are important for optimal operation of many industrial process units like the bubble column reactors. The population balance equation (PBE) is considered a concept for describing the evolution of populations of countable entities such as the bubbles in the bubble column. In recent studies, the least-squares method has been adopted for the solution of population balance (PB) problems. A favorable property of the weighted residual methods such as the least-squares technique is that the solution of the density function itself can be obtained from the fundamental PBE formulation, that is, not moment formulations. Hence, in this framework, the inn...

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