Physics: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. 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:

[math]\displaystyle{ d_s = \sqrt{\frac{A_p}{\pi}} }[/math]

and volume diameter, d_v:

[math]\displaystyle{ d_v = \left(\frac{6 V_p}{\pi}\right)^{1/3}, }[/math]

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 (SD) for a given particle is

[math]\displaystyle{ SD = D[3,2] = d_{32} = \frac{d_v^3}{d_s^2} }[/math]

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

[math]\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} }[/math]

[math]\displaystyle{ d_{32} = 6\frac{V_p}{A_p}. }[/math]

This is usually taken as the mean of several measurements, to obtain the Sauter mean diameter, SMD:

[math]\displaystyle{ SMD = \bar{d_{32}} = \sum_i\frac{d_v^3}{d_s^2} }[/math]

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.

[math]\displaystyle{ D_s = \frac{1}{\sum_i \frac{f_i}{d_i}} }[/math]

[math]\displaystyle{ f_i }[/math] is the scalar variable for the dispersed phase

[math]\displaystyle{ d_i }[/math] 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