Population balance equations (PBE) applied to comminution processes are commonly based on selection and breakage functions, which allow the description of changes of particle-size distributions vs. time. However, other properties, such as the particle strength, also influence the grinding kinetics. A bivariate PBE was developed and resolved by the direct quadrature method of moments. This equation includes both the particle size and strength, the latter of which is defined as the minimum energy required for breakage. The monovariate case was first validated by comparing the predicted moments with those calculated from the size distributions given by an analytical solution of the PBE derived for specific selection and breakage functions. The bivariate model was then compared with a discretized model to evaluate its validity. Finally, the benefit of the bivariate model was proven by analyzing the sensitivity of some parameters and comparing the results of the monovariate and bivariate cases. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1621–1631, 2014
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