On the existence of non-trivial steady-state size-distributions for a class of flocculation equations

TL;DR

This paper proves the existence of non-trivial steady-state size distributions in a generalized size-structured flocculation model, incorporating biological phenomena, and develops a numerical scheme to approximate these solutions for environmental and engineering applications.

Contribution

It provides a unified mathematical framework for various flocculation models and establishes conditions for positive steady states, including a spectral collocation numerical method.

Findings

Existence of positive steady states under certain conditions

Development of a spectral collocation numerical scheme

Insights into particle removal efficiency

Abstract

Flocculation is the process whereby particles (i.e., flocs) in suspension reversibly combine and separate. The process is widespread in soft matter and aerosol physics as well as environmental science and engineering. We consider a general size-structured flocculation model, which describes the evolution of floc size distribution in an aqueous environment. Our work provides a unified treatment for many size-structured models in the environmental, industrial, medical, and marine engineering literature. In particular, the mathematical model considered in this work accounts for basic biological phenomena in a population of microorganisms including growth, death, sedimentation, predation, surface erosion, renewal, fragmentation and aggregation. The central objective of this work is to prove existence of positive steady states of this generalized flocculation model. Using results from fixed point theory we derive conditions for the existence of continuous, non-trivial stationary solutions. We further develop a numerical scheme based on spectral collocation method to approximate these positive stationary solutions. We explore the stationary solutions of the model for various biologically relevant parameters and give valuable insights for the efficient removal of suspended particles.