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# Sauter diameter

The Sauter mean diameter (SMD) is defined as the diameter of a sphere that has the same volume/surface area ratio as a particle of interest. Originally developed by German scientist, J. Sauter in the late 1920s. The SMD is common measure in fluid dynamics as a way estimating the average particle size. Several methods have been devised to obtain a good estimate of the SMD. The SMD is typically defined in terms of the surface diameter, ds: $d_s = \sqrt{\frac{A_p}{\pi}}$

and volume diameter, dv: $d_v = \left(\frac{6 V_p}{\pi}\right)^{\frac{1}{3}}$

where Ap and Vp are the surface area and volume of the particle, respectively. ds and dv are usually measured directly by other means without knowledge of Ap or Vp. The Sauter diameter for a given particle is: $SD = D[3,2] = d_{32} = \frac{d_v^3}{d_s^2}$

If the actual surface area, Ap and volume, Vp of the particle are known the equation simplifies further: $\frac{V_p}{A_p} = \frac{\frac{4}{3}\pi (d_{32}/2)^3}{4\pi (d_{32}/2)^2} = \frac{(d_{32}/2)^3}{3 (d_{32}/2)^2} = \frac{d_{32}}{6}$ $d_{32} = 6\frac{V_p}{A_p}$

This usually taken as the mean of several measurement, to obtain the Sauter mean diameter, SMD: $SMD = \bar{d_{32}} = \sum_i\frac{d_v^3}{d_s^2}$

The provides intrinsic data that helps determine the particle size for fluid problems.

## Applications

The Sauter mean diameter (SMD) can be defined as the diameter of a drop having the same volume/surface area ratio as the entire spray. $D_{s} = \frac{1}{\sum_{i} \frac{f_{i}}{d_{i}}}$
fi is the scalar variable for the dispersed phase
di is the discrete bubble size

Sauter Mean Diameter (SMD) is also referred to as D[3,2]. It is especially important in calculations where the active surface area is important. Such areas include catalysis and applications in fuel combustion.