Sauter mean diameter

In fluid dynamics, Sauter mean diameter (SMD, d₃₂ or D[3, 2]) is an average of particle size. It was originally developed by German scientist Josef Sauter in the late 1920s.[1][2] It is defined as the diameter of a sphere that has the same volume/surface area ratio as a particle of interest. Several methods have been devised to obtain a good estimate of the SMD.

SMD is typically defined in terms of the surface diameter, d_s:

${\displaystyle d_{s}={\sqrt {\frac {A_{p}}{\pi }}}}$

and volume diameter, d_v:

${\displaystyle d_{v}=\left({\frac {6V_{p}}{\pi }}\right)^{1/3},}$

where A_p and V_p are the surface area and volume of the particle, respectively. If d_s and d_v are measured directly by other means without knowledge of A_p or V_p, Sauter diameter for a given particle is

${\displaystyle SD=D[3,2]=d_{32}={\frac {d_{v}^{3}}{d_{s}^{2}}}.}$

If the actual surface area, A_p and volume, V_p of the particle are known the equation simplifies further:

${\displaystyle {\frac {V_{p}}{A_{p}}}={\frac {{\frac {4}{3}}\pi (d_{v}/2)^{3}}{4\pi (d_{s}/2)^{2}}}={\frac {(d_{v}/2)^{3}}{3(d_{s}/2)^{2}}}={\frac {d_{32}}{6}}}$

${\displaystyle d_{32}=6{\frac {V_{p}}{A_{p}}}.}$

This is usually taken as the mean of several measurements, to obtain the Sauter mean diameter, SMD: This provides intrinsic data that help determine the particle size for fluid problems.

Applications

The SMD can be defined as the diameter of a drop having the same volume/surface area ratio as the entire spray.

${\displaystyle D_{s}={\frac {1}{\sum _{i}{\frac {f_{i}}{d_{i}}}}}}$

${\displaystyle f_{i}}$ is the scalar variable for the dispersed phase

${\displaystyle d_{i}}$ is the discrete bubble size

SMD is especially important in calculations where the active surface area is important. Such areas include catalysis and applications in fuel combustion.

See also

• Sphericity

References

1. Sauter J. "Die Grössenbestimmung der in Gemischnebeln von Verbrennungskraftmaschinen vorhandenen Brennstoffteilchen" VDI-Forschungsheft Nr. 279 (1926) und Nr. 312 (1928) ISSN 0042-174X
2. Wang, D.; Fan, L.-S. (2013). "Particle characterization and behavior relevant to fluidized bed combustion and gasification systems". In Scala, Fabrizio (ed.). Fluidized Bed Technologies for Near-Zero Emission Combustion and Gasification. Elsevier. pp. 42–76 [p. 45]. doi:10.1533/9780857098801.1.42.