Exact solutions of temperature-dependent Smoluchowski equations

TL;DR

This paper presents exact solutions to temperature-dependent Smoluchowski equations, providing benchmarks for numerical methods and insights into different evolution regimes of agglomerates with temperature effects.

Contribution

It introduces a set of exact solutions for specific model rate coefficients, enhancing understanding and validation of numerical approaches for these equations.

Findings

Exact solutions serve as benchmarks for numerical methods.

Excellent agreement between numerical and exact solutions.

Infinite model coefficients allow for exact analysis.

Abstract

We report a number of exact solutions for temperature-dependent Smoluchowski equations. These equations quantify the ballistic agglomeration, where the evolution of densities of agglomerates of different size is entangled with the evolution of the mean kinetic energy (partial temperatures) of such clusters. The obtained exact solutions may be used as a benchmark to assess the accuracy and computational efficiency of the numerical approaches, developed to solve the temperature-dependent Smoluchowski equations. Moreover, they may also illustrate the possible evolution regimes in these systems. The exact solutions have been obtained for a series of model rate coefficients, and we demonstrate that there may be an infinite number of such model coefficient which allow exact analysis. We compare our exact solutions with the numerical solutions for various evolution regimes; an excellent agreement between numerical and exact results proves the accuracy of the exploited numerical method.