# Convenient analytical formula for cluster mean diameter and diameter dispersion after nucleation burst

## TL;DR

This paper introduces an analytical formula to estimate the mean diameter and dispersion of clusters immediately after nucleation in a cooling gas, simplifying predictions of cluster properties based on process parameters.

## Contribution

It derives explicit formulas for cluster diameter and dispersion using a moment model, enabling easier prediction and input for subsequent coagulation modeling.

## Key findings

- Formulas agree well with numerical simulations
- Predicts diameter and dispersion based on cooling and collision times
- Facilitates input for coagulation stage modeling

## Abstract

We propose a new method of estimating the mean diameter and dispersion of clusters formed in a cooling gas, right after the nucleation stage. Using a moment model developed by Friedlander [S.K. Friedlander, Ann. N.Y. Acad. Sci. 354 (1983)], we derive an analytic relationship for both cluster diameter and diameter dispersion as a function of two of the characteristic times of the system - the cooling time and primary constituents collision time. These formulas can be used to predict diameter and dispersion variation with process parameters such as the initial monomer pressure or cooling rate. It is also possible to use them as an input to the coagulation stage, without the need to compute complex cluster generation during the nucleation burst. We compared our results with a nodal code and got excellent agreement.