Log-normal Distribution
Aerosol size distributions typically follow log-normal distributions, which is determined by the physical mechanisms of particle formation and growth.
Mathematical Definition
Probability Density Function
\[\frac{dN}{d\ln D_p} = \frac{N_t}{\sqrt{2\pi}\ln\sigma_g} \exp\left[-\frac{(\ln D_p - \ln D_{pg})^2}{2\ln^2\sigma_g}\right]\]
Where: - \(N_t\) = Total number concentration - \(D_{pg}\) = Geometric mean diameter (GMD) - \(\sigma_g\) = Geometric standard deviation (GSD)
Common Representations
| Representation | Symbol | Unit | Description |
|---|---|---|---|
| dN | \(dN\) | #/cm3 | Number concentration |
| dN/dDp | \(dN/dD_p\) | #/cm3/nm | Number per diameter |
| dN/dlogDp | \(dN/d\log D_p\) | #/cm3 | Number per log diameter |
Distribution Conversion
Number to Surface Area
\[\frac{dS}{d\log D_p} = \pi D_p^2 \cdot \frac{dN}{d\log D_p}\]
Number to Volume
\[\frac{dV}{d\log D_p} = \frac{\pi}{6} D_p^3 \cdot \frac{dN}{d\log D_p}\]
Hatch-Choate Conversion
Relationship between GMD of different weightings:
\[\ln D_{pg,S} = \ln D_{pg,N} + 2\ln^2\sigma_g$$
$$\ln D_{pg,V} = \ln D_{pg,N} + 3\ln^2\sigma_g\]
Modal Classification
Atmospheric aerosols typically contain multiple modes:
| Mode | Size Range | Primary Sources |
|---|---|---|
| Nucleation | 1-25 nm | Gas-to-particle conversion, new particle formation |
| Aitken | 25-100 nm | Growth, combustion emissions |
| Accumulation | 100-1000 nm | Aging, cloud processing |
| Coarse | >1000 nm | Mechanical processes, sea salt, dust |
Statistical Calculations
Geometric Mean Diameter (GMD)
\[D_{pg} = \exp\left(\frac{\sum n_i \ln D_{p,i}}{\sum n_i}\right)\]
Geometric Standard Deviation (GSD)
\[\ln\sigma_g = \sqrt{\frac{\sum n_i (\ln D_{p,i} - \ln D_{pg})^2}{\sum n_i}}\]
Mode Diameter
The diameter corresponding to the distribution peak.
In AeroViz
Usage lives in the guide: Distribution Conversion and Statistics covers SizeDist.properties() / mode_statistics() and the psd_stats shortcut.
Multi-modal Fitting
Log-normal mixture model:
\[\frac{dN}{d\log D_p} = \sum_{i=1}^{n} \frac{N_i}{\sqrt{2\pi}\log\sigma_{g,i}} \exp\left[-\frac{(\log D_p - \log D_{pg,i})^2}{2\log^2\sigma_{g,i}}\right]\]
References
- Seinfeld, J. H., & Pandis, S. N. (2016). Atmospheric Chemistry and Physics: From Air Pollution to Climate Change. Wiley.
- Hinds, W. C. (1999). Aerosol Technology: Properties, Behavior, and Measurement of Airborne Particles. Wiley.