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    Item type:Publication, Access status: with files ,
    Advanced approximation method with a convergence acceleration parameter for multidimensional aggregation model
    (Elsevier, 2026-08-03)
    Yadav, Sonia
    ;
    Singh, Akashdeep
    ;
    Singh, Sukhjit
    ;
    Walker, Gavin
    ;
    Heinrich, Stefan
    This study presents a highly efficient and accurate approximate method based on a convergence acceleration parameter to approximate a nonlinear multidimensional aggregation population balance equation. Optimal tuning of the acceleration parameter significantly enhances solution quality over extended temporal domains and overcomes key limitations of existing approaches. Deeper mathematical insight is provided through a discussion of the existence of the proposed approach within the framework of a nonlinear aggregation model. Convergence analysis and error estimates are established using the fixed point theorem and the contractive mapping principle, thereby proving the existence of solutions to the aggregation model. The accuracy and efficiency of the proposed approach are demonstrated by computing approximate solutions for the number density function and its moments for physically relevant kernels. For analytically tractable kernels, results are validated against exact solutions. For complex size-dependent kernels, including polymerization, Ruckenstein–Pulvermacher, and shear kernels, the obtained results are compared with the existing finite volume scheme, homotopy analysis method, and optimal decomposition method. The results show that the proposed approach achieves higher accuracy in capturing number density functions and their integral moments while requiring significantly fewer series terms than existing methods.
    Publicationtype: Journal Article
      22  8
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    Item type:Publication, Access status: with files ,
    Deterministic methodologies for aggregation, breakage, growth and nucleation population balances
    (Springer, 2026-03-10)
    Ansari, Zeeshan
    ;
    Rae, Mitchell
    ;
    Sriwastav, Nikhil
    ;
    Dewangan, Mainendra
    ;
    Tomar, Saurabh
    Population balance equations (PBEs) have previously been applied for modelling across different fields including astrophysics for cloud formation, bubble dynamics, and liquid-liquid dispersion processes. In this paper, a generalised numerical method is presented by enabling the efficient solution of aggregation, breakage, growth, and nucleation mechanisms. A volume-conserving finite volume approximation is developed for the aggregation and breakage processes while the method of characteristics is used for the growth process. This addresses the instability in solutions of combined breakage-growth or aggregation-growth PBEs. Additionally, a nuclei-size cell is introduced at each time step to handle the combined aggregation-nucleation and breakage-nucleation processes. This approach eliminates the need to convert the original aggregation and breakage population balances into divergence form, allowing stable numerical solutions to be obtained for combined processes [Math. Models Methods Appl. Sci. 23(7), 235–1273]. The new numerical approximations are robust for handling complex simultaneous processes, effectively reducing numerical diffusion and dispersion. The accuracy and efficiency of the proposed scheme are validated through comparisons with newly derived analytical results and existing finite volume schemes, fixed pivot technique, and cell average technique, demonstrating its superiority in performance. The comparison shows that the new schemes predict number density functions (NDFs) and their integral moments with significantly higher precision. Computationally, the proposed approach captured all numerical results by consuming 40–70% lesser computational time compared to existing schemes.
    Publicationtype: Journal Article
      47  18
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    Item type:Publication, Access status: without files ,
    An accurate approach and its convergence analysis for the multidimensional nonlinear collisional breakage equations
    (Elsevier, 2026-01-29)
    Arora, Gourav
    ;
    Hussain, Saddam
    ;
    Walker, Gavin
    ;
    Cattani, Carlo
    ;
    Heinrich, Stefan
    Particulate processes such as bubble breakage, granule preparation, and ore extraction are often modeled using population balance equations. While most studies rely on linear fragmentation models, these fail to capture realistic dynamics where breakage occurs due to particle collisions, leading to nonlinear collisional breakage equations. The mathematical complexity of nonlinear kernels and integrals has limited progress in this direction, with very few analytical solutions available. In this work, we develop an accurate approach based on a modified variational iteration method for solving multidimensional collisional breakage equations. The method allows flexible operator selection and efficient determination of the Lagrange multiplier. A rigorous convergence and error analysis is carried out using the fixed-point theorem. The approach is further extended to two- and three-dimensional models, including collisional fragmentation cases. Numerical experiments confirm the accuracy and efficiency of the method compared with existing approaches.
    Publicationtype: Journal Article
      11