A comparative study of the fixed pivot technique and finite volume schemes for multi-dimensional breakage population balances

When modeling particle breakage, the use of a single dimension to characterize particulate systems becomes intractable when multiple dimensions (e.g., size and shape) are important. Amongst many potential numerical techniques, the Fixed Pivot Technique (FPT) and Finite Volume Schemes (FVS) are popular choices to resolve multi-dimensional breakage population balance equations (PBEs). However, whether there exists a general multi-purpose technique between the two remains unclear. Across all test cases with identical size domain and mesh, while both techniques demonstrate comparable accuracy in resolving the moments and number densities (with maximum difference in average relative errors of ∼O(10⁰) and ∼O(10²), respectively), and exhibit similar computational efficiency (time taken by FPT relative to FVS is of ∼O(10⁰)), the choice of numerical technique is contingent upon the properties where accurate prediction is critical. To this end, FVS is the preferred choice when precise estimation of up to two properties is required owing to its simplicity, albeit requiring distinct schemes for different properties. Conversely, when more than two properties are crucial, the FPT is more suited as it preserves up to four properties in the internal 2D space. Overall, this work offers rational guidance for efficient and accurate modeling of multi-dimensional breakages.