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Volume 103
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Shang, C., He, S., Xu, Z., Kruis, F. E., & Zhao, H. (2025). Parameter identification in the bismuth homogeneous nucleation model for population balance. Particuology, 103, 117-127. https://doi.org/10.1016/j.partic.2025.05.012
Parameter identification in the bismuth homogeneous nucleation model for population balance
Cheng Shang a, Song He a, Zuwei Xu a, Frank Einar Kruis b, Haibo Zhao a *
a State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
b Faculty of Engineering Sciences, Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen, Bismarckstr. 81, 47057 Duisburg, Germany
10.1016/j.partic.2025.05.012
Volume 103, August 2025, Pages 117-127
Received 23 October 2024, Revised 24 April 2025, Accepted 12 May 2025, Available online 28 May 2025, Version of Record 6 June 2025.
E-mail: hzhao@mail.hust.edu.cn
Highlights

• Surface tension of bismuth metal is identified as 0.466 N/m in homogenous nucleation simulation.

• Spatiotemporal Monte Carlo method is used in the direct problem and the response surface method for inverse problems.

• 17 sets of parameters for three different mechanisms are used to construct the response surface model.

• Comprehensive sum of mean square error is used to determine the most appropriate parameters.


Abstract

A new multivariate population balance modeling for the homogenous nucleation from bismuth vapor is tested against experimental results. However, the small differences of some key parameters could lead to the ill-posed problem, such as the surface tension. In this study, the parameter fluctuations set according to the response surface method are employed in the population balance simulation to identify their importance. Subsequently, the quadratic polynomials are established to replace the simulation and the fluctuations are evaluated with the characteristic parameters of particle evolution. It is found that the surface tension tends to be the most significant factor determining the particle evolution, which is also influenced by the coefficient in condensation rates and fractal dimension in the coagulation. To get more accurate modeling and key parameters, the comprehensive sum of mean square error is calculated based on all the fluctuations and the appropriate value for the surface tension is 0.466 N/m.

Graphical abstract
Keywords
Population balance modeling; Bismuth; Homogeneous nucleation; Response surface method; Parameter identification
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