Particle size statistics: the Sauter mean diameter (often used in mass transfer) is identical to which named mean?

Introduction / Context:
Different particle size "means" emphasize different physical phenomena. The Sauter mean diameter d32 is central in interfacial processes (e.g., droplet evaporation, gas–liquid mass transfer) because it weights particles by surface area per unit volume. Recognizing its equivalence to the volume–surface mean helps in choosing the right metric for design correlations.

Given Data / Assumptions:

• A polydisperse particle or droplet population.
• Interest in surface-area-related transport.
• Standard nomenclature for statistical means.

Concept / Approach:
The Sauter mean diameter d32 is defined so that the total surface area-to-volume ratio of the real distribution equals 6/d32 (spherical assumption). Algebraically, d32 = Σ(n_i d_i^3)/Σ(n_i d_i^2), the ratio of the third to the second moment of size distribution. This is exactly the volume-surface mean because the numerator represents a volume-type weighting and the denominator a surface-type weighting.

Step-by-Step Solution:
Recall definition: d32 emphasizes particles by contribution to surface area for a given volume.Match to named mean: volume–surface mean is the same construction.Select "volume-surface" as the correct option.

Verification / Alternative check:
Mass transfer correlations (e.g., k_L a) often employ d32 because a ∝ 6/d32 for spherical dispersions, directly linking size to interfacial area per unit volume.

Why Other Options Are Wrong:
Arithmetic, geometric, and mass means do not preserve surface-to-volume equivalence and would mispredict interfacial area for polydisperse systems.

Common Pitfalls:
Confusing number mean or volume mean with Sauter mean; using the wrong mean can produce large errors in predicted rates.

Final Answer:
volume-surface