Ratio of the population averaged Knudsen numbers with the monodisperse Knudsen number. Due to the gamma function properties, a pole occurs at values of ν ≤ 1/3 for the number-weighted Knudsen number (blue solid line), leading to singularities at ν ≤ 1/3. The incomplete gamma function formulation using Eq. (A.11), which accounts for a cutoff at small particle sizes, avoids the singularities in the gamma function (dashed blue line) and is valid across the ν range.