An analytical expression for the mobility diameter of fractal aggregates valid for a wide range of temperatures and its comparison with popular laws

The size measurement of airborne nanoparticles is primarily performed using commercial devices that classify particles based on their electrical mobility. This method relies on balancing the hydrodynamic drag force induced by particle entrainment in a laminar airflow with a transverse electric force. For spherical particles, this mobility-based selection directly correlates with their geometric diameter. However, for non-spherical particles, this relationship is less straightforward. To address this complexity, shape factors have long been introduced to estimate the diameter of volume-equivalent spherical particles from their mobility. Yet, aerosol nanoparticles are frequently encountered as fractal aggregates formed by nearly spherical primary spheres, a morphological configuration that has been mathematically characterized using the scaling law. Extensive research has sought to establish accurate relationships between mobility diameter and the morphological descriptors of these aggregates, such as the number and radii of primary spheres, fractal dimension, and prefactor. While this topic has inspired prolific research, it remains an active field of investigation due to significant discrepancies among available analytical models which are useful for interpreting size distributions or for improving the modeling of the mobility of fractal aggregates in codes such as the Population Balance equation. This paper focuses on the most advanced and commonly used analytical expressions, casting them within a unified and simplified analytical framework to discuss their differences and similarities in terms of the role of primary sphere Knudsen number. This approach facilitates the comparison of different models and enhances the understanding of the underlying physical significance. Finally, we propose a simple analytical expression for the mobility diameter of fractal aggregates valid across a wide range of morphologies, temperatures, and flow regimes. The effect of primary sphere overlap and polydispersity is also examined. This model is readily implementable in any post-processing code, offering practical utility for real-world applications.