(87c) Mass Conservative Solution to the Dispersed Phase Size Distribution Using the Least-Squares Spectral Elements Method

Loss of mass of dispersed phase in the standard least-squares formulation of the population balance equation is observed. This loss of mass is caused by inexact conservation property of some breakage models, which might result in incorrect physical interpretations.

Investigations on the reason for the poor mass conservation quality of some breakage models has not yet been reported. To the authors' knowledge, no remedy or modification in attempting to fix this critical problem has been made, either. This points out the need to find a possible way that improves those breakage models' performances by a guaranteed mass conservation.

In this work a constraint least-squares spectral element method is developed which enforces the mass conservation. This is accomplished by adding an extra restriction in the dispersed phase continuity equation through the Lagrange multipliers strategy. This modified version of least-squares formulation has been tested and compared with the standard one in to simulate the two-phase flow passing through a 2D domain, and the Martinez-Bazan breakage kernel is employed in the population balance equation.

The discretized system resulting from applying the method to a two-phase population balance equation problem is symmetric and pesudopositive definite. Results obtained by the modified least-squares spectral element method show that the mass is conserved everywhere with high accuracy.