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Semi-Analytical Methods for Population Balance models involving Aggregation and Breakage processes: A comparative study

Shweta, Saddam Hussain, Rajesh Kumar

TL;DR

The paper addresses nonlinear population balance equations (PBEs) governing aggregation and aggregation-breakage (CABE) by introducing an Accelerated Homotopy Analysis Method (AHAM). It presents a rigorous HAM framework, augments it with an acceleration scheme, and provides contraction-based convergence proofs in Banach spaces for both pure aggregation and CABE. Through extensive numerical tests across multiple coagulation kernels, AHAM shows high accuracy and outperforms existing semi-analytical methods (ADM, HPM, HAM, ODM, LODM) in stability and convergence of density functions $c(s,\tau)$ and their moments. The work delivers a robust, fast semi-analytical tool for PBEs with potential impact on aerosol physics, crystallization, pharmaceutical processing, and related particle dynamics applications, with clear pathways for extension to other nonlinear PBEs.

Abstract

Population balance models often integrate fundamental kernels, including sum, gelling and Brownian aggregation kernels. These kernels have demonstrated extensive utility across various disciplines such as aerosol physics, chemical engineering, astrophysics, pharmaceutical sciences and mathematical biology for the purpose of elucidating particle dynamics. The objective of this study is to refine the semi-analytical solutions derived from current methodologies in addressing the nonlinear aggregation and coupled aggregation-breakage population balance equation. This work presents a unique semi-analytical approach based on the homotopy analysis method (HAM) to solve pure aggregation and couple aggregation-fragmentation population balance equations, which is an integro-partial differentia equation. By decomposing the non-linear operator, we investigate how to utilize the convergence control parameter to expedite the convergence of the HAM solution towards its precise values in the proposed method.

Semi-Analytical Methods for Population Balance models involving Aggregation and Breakage processes: A comparative study

TL;DR

The paper addresses nonlinear population balance equations (PBEs) governing aggregation and aggregation-breakage (CABE) by introducing an Accelerated Homotopy Analysis Method (AHAM). It presents a rigorous HAM framework, augments it with an acceleration scheme, and provides contraction-based convergence proofs in Banach spaces for both pure aggregation and CABE. Through extensive numerical tests across multiple coagulation kernels, AHAM shows high accuracy and outperforms existing semi-analytical methods (ADM, HPM, HAM, ODM, LODM) in stability and convergence of density functions and their moments. The work delivers a robust, fast semi-analytical tool for PBEs with potential impact on aerosol physics, crystallization, pharmaceutical processing, and related particle dynamics applications, with clear pathways for extension to other nonlinear PBEs.

Abstract

Population balance models often integrate fundamental kernels, including sum, gelling and Brownian aggregation kernels. These kernels have demonstrated extensive utility across various disciplines such as aerosol physics, chemical engineering, astrophysics, pharmaceutical sciences and mathematical biology for the purpose of elucidating particle dynamics. The objective of this study is to refine the semi-analytical solutions derived from current methodologies in addressing the nonlinear aggregation and coupled aggregation-breakage population balance equation. This work presents a unique semi-analytical approach based on the homotopy analysis method (HAM) to solve pure aggregation and couple aggregation-fragmentation population balance equations, which is an integro-partial differentia equation. By decomposing the non-linear operator, we investigate how to utilize the convergence control parameter to expedite the convergence of the HAM solution towards its precise values in the proposed method.
Paper Structure (14 sections, 4 theorems, 84 equations, 15 figures, 2 tables)

This paper contains 14 sections, 4 theorems, 84 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. State-of-the Art and Contribution
  3. Motivation
  4. Methodology
  5. HAM
  6. Accelerated Homotopy Analysis Method
  7. Development of the method for Aggregation and coupled aggregation-fragmentation Equations
  8. AHAM Implementation for Pure Aggregation
  9. Convergence Analysis for Pure Aggregation Equation
  10. AHAM implementation for coupled aggregation-breakage equation (CABE)
  11. Convergence analysis for CABE
  12. Numerical Testing
  13. Coupled Aggregation-Breakage Equation (CABE)
  14. Concluding Remarks

Key Result

Theorem 3.1

The operator $\widetilde{\mathcal{A}}$ is contractive on $\mathbb{X}_{1}$, i.e; $\|\widetilde{\mathcal{A}}c-\widetilde{\mathcal{A}}c^{\ast}\| \leq \gamma \|c-c^{\ast}\|$, $\forall$$c,c^{\ast}\in\mathbb{X}_{1}$ with

Figures (15)

  • Figure 1: Number density and error plots for Example \ref{['q1']}
  • Figure 2: Absolute errors for Example \ref{['q1']}
  • Figure 3: Zeroth and second moments for Example \ref{['q1']}
  • Figure 4: Number density and error plots for Example \ref{['q2']}
  • Figure 5: AHAM, HAM and ODM errors for Example \ref{['q2']}
  • ...and 10 more figures

Theorems & Definitions (18)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 8 more