# Stochastic & Numerical Algorithm Development

Computational techniques are the foundation of all the work done within the Computational Modelling Group. Indeed, many of the physical systems investigated by various members of the group were intractable before the advent of modern computer hardware and efficient numerical algorithms. Besides using the latest computational techniques, many members of the group take an active interest in the mathematical development, formulation and practical implementation of *new* algorithms within the various fields of interest.

As an example, note that the particle dynamics of many processes of interest (for example, in the Nanoparticles and Particle Processes themes), are governed by a population balance. The simplest form of such a population balance is the Smoluchowski coagulation equation. A more detailed - and physically useful - model is formed by adding additional terms to this equation to account for new particles entering the system through formation in the gas phase, surface growth through contact with gaseous species and coalescence through sintering. The governing equation thus modified is a bivariate population balance with volume and surface area as the two internal coordinates. This can be solved to give particle size distributions at various locations in, for example, a flame.

Bivariate population balances are computationally intensive to solve using standard numerical techniques. To overcome this obstacle, the group has developed a Stochastic Particle Algorithm for solving such equations. This represents a breakthrough in numerical efficiency for solving population balances. The efficiency of the algorithm has enabled the simulation of a greater number of internal coordinates: in the extreme case it is now possible to simulate the full spatial structure of the agglomerates. This allows the visualisation of the simulated particles and subsequent direct comparison with TEM micrographs.

## Recent Associated Preprints

299: On the role of C_{4} and C_{5} products in electrochemical CO_{2} reduction

Simon D. Rihm, Mikhail K. Kovalev, Alexei A. Lapkin, Joel W. Ager, and Markus Kraft, Technical Report 299, c4e-Preprint Series, Cambridge, 2022.

289: Modelling a detailed kinetic mechanism for electrocatalytic reduction of CO_{2}

Simon D. Rihm, Jethro Akroyd, and Markus Kraft, Technical Report 289, c4e-Preprint Series, Cambridge, 2022.

278: Detailed modelling of aerosol growth dynamics

Astrid Boje and Markus Kraft, Technical Report 278, c4e-Preprint Series, Cambridge, 2021.

268: Predicting power conversion efficiency of organic photovoltaics: models and data analysis

Andreas Eibeck, Daniel Nurkowski, Angiras Menon, Jiaru Bai, Jinkui Wu, Li Zhou, Sebastian Mosbach, Jethro Akroyd, and Markus Kraft, Technical Report 268, c4e-Preprint Series, Cambridge, 2021.

## Recent Associated Publications

On the role of C_{4} and C_{5} products in electrochemical CO_{2} reduction

Simon D. Rihm, Mikhail K. Kovalev, Alexei A. Lapkin, Joel W. Ager, and Markus Kraft, Energy & Environmental Science 16(4), 1697-1710, (2023).

Stochastic population balance methods for detailed modelling of flame-made aerosol particles

Astrid Boje and Markus Kraft, Journal of Aerosol Science 159, 105895, (2022).

Predicting Power Conversion Efficiency of Organic Photovoltaics: Models and Data Analysis

Andreas Eibeck, Daniel Nurkowski, Angiras Menon, Jiaru Bai, Jinkui Wu, Li Zhou, Sebastian Mosbach, Jethro Akroyd, and Markus Kraft, ACS Omega 6(37), 23764-23775, (2021).

Jinkui Wu, Shihui Wang, Li Zhou, Xu Ji, Yiyang Dai, Yagu Dang, and Markus Kraft, Industrial & Engineering Chemistry Research 59(42), 18991-19000, (2020).

## Funding

Funding has generously been provided by EPSRC, Toyota, CMCL Innovations, and CREATE.